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Accession  No.  -  #          Class  No. 


THE   ELEMENTS   OF   PHYSICS 

VOL.   Ill 
LIGHT   AND   SOUND 


THE 


ELEMENTS    OF    PHYSICS 


A  COLLEGE   TEXT-BOOK 


BY 

EDWARD    L.   NICHOLS 

AND 

WILLIAM    S.    FRANKLIN 


IN  THREE    VOLUMES 

VOL.  Ill 

LIGHT  AND  SOUND 


gork 
THE   MACMILLAN    COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 
1897 

All  rights  reserved 


Engineering 
JUbiary 


COPYKIGHT,  1897, 

BY  THE  MACMILLAN  COMPANY. 


Norton  oti 

J.  S.  Cashing  &  Co.  -  Berwick  8s  Smith 
Norwoqd  Mass.  U.S.A. 


TABLE   OF   CONTENTS. 


CHAPTER   I. 

PAGE 

Light  and  sound  defined ;  Measurement  of  velocities  i 

CHAPTER   II. 

Longitudinal  and  transverse  waves  ;  Equations  of  wave  motion ;  Wave 
trains,  simple  and  compound  ;  Fourier's  theorem ;  The  wave  front ; 
Shadows  ...........  9 

CHAPTER   III. 

Reflection  and  refraction ;  Mirrors  and  the  formation  of  images ;  Re- 
fraction of  plane  and  spherical  waves  ;  Total  reflection  ;  Spherical 
aberration .  26 


CHAPTER   IV. 

Lenses  and  lens  systems ;    Specification  of  a  system ;    Principal  foci, 

principal  planes,  and  nodal  points 45 


CHAPTER   V. 

The  correction  of  lenses  and  lens  systems ;  Spherical  and  chromatic 
aberration  ;  Astigmatism  ;  Distortion  ;  Curvature  of  field  ;  Descrip- 
tion of  lens  systems  used  in  practice  .  .  .  .  .  -55 


CHAPTER  VI. 

The  eye;    The  photographic  camera;    The  projecting  lantern;    The 

microscope  ;  The  telescope  ;  Measurement  of  magnifying  power      .       64 


vi  CONTENTS. 

CHAPTER   VII. 

PAGE 

Newton's  experiment ;  The  spectrum ;  The  achromatic  lens  ;  The  spec- 
troscope ;  Classes  of  spectra ;  The  spectrometer ;  The  spectro- 
photometer 73 

CHAPTER   VIII. 

Interference  from  similar  sources  ;  Arrangements  for  producing  fringes  ; 
Colors  of  thin  plates ;  Newton's  rings ;  Diffraction  past  an  edge ; 
Diffraction  through  a  slit ;  Zone  plates ;  The  diffraction  grating ; 
The  grating  spectrometer ;  The  measurement  of  wave  lengths  .  82 

CHAPTER   IX. 

Sensations  of  brightness  and  color ;  Luminosity ;  Colors  due  to  homo- 
geneous and  to  mixed  light ;  Color  mixing ;  Dichroic  vision  ;  Test- 
ing for  color  blindness 101 

CHAPTER   X. 

Photometry ;  The  law  of  inverse  squares  ;  Standards  of  light ;  Bouguer's 
principle  ;  Simple  photometers  ;  Distribution  of  brightness ;  The 
photometry  of  lights  differing  in  composition 112 

CHAPTER   XI. 

Polarization  defined ;  Behavior  of  tourmaline  ;  Polarization  by  reflection ; 
Double  refraction  ;  The  Nicol  prism  ;  The  polariscope ;  Rotation  of 
the  plane  of  polarization 125 


CHAPTER  XII. 

Radiant  heat ;  Prevost's  principle  ;  Law  of  normal  radiation  ;  Selective 
emission,  reflection  and  transmission  ;  Selective  absorption  ;  Black 
bodies  ;  White  bodies  ;  Surface  color ;  Methods  of  measuring  radi- 
ant heat;  Fluorescence  and  Phosphorescence 135 


CHAPTER   XIII. 

Vibration   of  a  particle ;    Simple  and  compound  vibrations ;    Musical 

tones  and  noises  ;  Loudness  ;  Pitch  ;  Timbre  .         .         .         .     147 


CONTENTS.  vii 

CHAPTER   XIV. 

PAGE' 

Vibrations  of  air  columns ;  Organ  pipes ;  The  clarionet ;  The  cornet ; 
The  vocal  organs  ;  Vibrations  of  rods  and  strings  ;  Kundt's  experi- 
ment ;  The  compound  vibration  of  strings ;  Chladni's  figures ;  The 
tuning  fork ;  Manometric  flames  .  .  .  .  .  .  .154 

CHAPTER  XV. 

Proper  and  impressed  vibrations ;   Damping ;  Resonance ;  Analysis  of 

clang;  Vowel  sounds ;  Reproduction  of  speech       .         .         .         .167 

CHAPTER   XVI. 

The  ear;  Persistence  of  sound  sensations  ;  Interference  of  sound  sensa- 
tions ;  Combination  tones  ;  The  echo  ;  Influence  of  diffraction  upon 
the  sense  of  direction  ;  Changes  of  pitch  due  to  relative  motions  .  173 

CHAPTER  XVII. 

Pitch  intervals;  Complete  and  approximate  consonance;  Major  and 
minor  accords  ;  Musical  scales  ;  Expression  ;  Rhythm,  melody,  har- 
mony, and  modulation 181 


HTY 


THE  ELEMENTS  OF  PHYSICS. 


VOLUME  III. 


CHAPTER   I. 
LIGHT   AND   SOUND   DEFINED;    VELOCITY. 

604.  Sensory  nerves. — The   sensory   nerves  of   the  human 
body  lead  from  regions  near  the  surface  of  the  body  to  the 
central  organs  of  the  nervous  system.     The  outer  ends  of  these 
nerves  are  exposed  in  such  a  way  as  to  be  excited  or  set  into 
commotion  by  physical  disturbances  in  the  region  surrounding 
the  body.     This  commotion  is  transmitted  to  the  central  organs 
producing  commotion  there,  and  we  perceive  what  we  call  a 
sensation.     The  physical  disturbance  which  excites  a  nerve  is 
called  a  stimulus. 

605.  Proper  stimuli.   End  organs  ;  localization. — Those  physi- 
cal disturbances  to  which  a  set  of  sensory  nerves  are  especially 
sensitive  are  called  the  proper  stimuli  of  that  set  of  nerves.     A 
set   of   sensory  nerves  is  rendered  especially  sensitive  to  its 
proper  stimuli  by  being  provided  with  terminal  or  end  organs 
which  are  easily  affected  by  those  stimuli,  and  by  being  so 
located  as  to   be   very   largely   protected   by  the   surrounding 
tissues  of  the  body  from  other  excitation. 

606.  Optic  nerves.      Light,   the  sensation;   light,  the  proper 
stimuhis.  —  The  end  organs  (rods  and  cones)  of  the  nerves  of 


2  ELEMENTS   OF   PHYSICS. 

sight  are  situated  in  the  retina  of  the  eye.  They  are  well  pro- 
tected by  the  surrounding  bones  from  all  physical  disturbances, 
except  such  as  can  reach  them  through  the  transparent  humors 
of  the  eye.  Severe  mechanical  shocks  and  electric  disturbances 
(electric  currents)  do,  however,  penetrate  to  these  end  organs 
and  excite  them.  The  sensation  which  is  perceived  when  the 
optic  nerves  are  excited  is  called  light.  That  physical  disturb- 
ance which  constitutes  the  proper  stimulus  of  the  optic  nerves 
is  also  called  light. 

607.  Auditory  nerves.    Sound,  the  sensation;  sound,  the  proper 
stimulus, — The  end  organs  of  the  nerves  of  hearing  are  situ- 
ated in  the  inner  ear.     They  are  well  protected  by  the  massive 
bones  of  the  head  from  all  physical  disturbances,  except  such 
as  can  travel  along  the  chain  of  tiny  bones  which  bridges  the 
cavity  between  the  tympanic  membrane  and  the  window  of  the 
inner  ear.     Severe  mechanical  shocks  and  electric  currents  do 
sometimes  penetrate  to  these  end  organs  through  the  surround- 
ing bones  and  excite  them.     The  sensation  which  is  perceived 
when  the  auditory  nerves  are  excited  is  called   sound.     That 
class    of    physical  disturbances   which    constitutes   the   proper 
stimulus  of  the  auditory  nerves  is  also  called  sound. 

608.  The  long-range  aspect  of  the  sensations  of  sight  and 
hearing.  —  We  have  come  by  experience  to  associate  more  or 
less  remote  objects  with  our  sensations  of  sight  and  hearing. 
A  physical  disturbance  reaches  our  eyes  from  an  object  which  we 
see  (or  our  ears  from  an  object  which  we  hear),  and  this  disturb- 
ance when  it  reaches  us  is  indicative  of  such  of  the  characteristics 
of  the  object  as  we  can  perceive  by  sight  (or  hearing}. 

609.  The  corpuscular  theory  of  light.  —  The  phenomena  of 
shadows   and  the  obstruction  of  vision  of  a  distant  object  by 
intervening  objects  show  that  light  travels  sensibly  in  straight 
lines.     In  accordance  with  this  fact,  it  was  the  accepted  theory, 
until  long  after  the  time  of  Sir  Isaac  Newton,  that  light  con- 


LIGHT   AND    SOUND   DEFINED.  3 

sisted  of  particles  or  corpuscles  which  were  thrown  off  from 
luminous  bodies  at  great  velocity,  traveling  in  straight  lines 
until  reflected  (or  stopped)  by  objects  upon  which  they  might 
impinge.  This  was  called  the  corpuscular  theory  of  light. 

610.  The  wave  theory  of  light  and  of  sound.  — The  most  com- 
prehensive understanding  of  the  phenomena  of  light  and  sound 
is  reached  if  we  look  upon  them  as  wave-like  disturbances  which 
pass  out  in  all  directions  from  luminous  and  from  sonorous 
bodies  respectively. 

The  assumption  that  light  and  sound  are  wave  motions  is 
verified  by  widest  experience.  No  attempt  will  be  made  to 
establish  this  assumption  by  any  preliminary  discussion.  The 
justification  of  the  wave  theory,  in  the  reader's  mind,  will 
become  more  and  more  complete  as  he  has  occasion  to  use  it. 

611.  The  luminiferous  ether.  —  The  conception  of  light  as  a 
wave-like  disturbance  depends  upon  the  assumed  existence  of  an 
all-pervading  medium,  —  the  ether.   The  fact  that  light  reaches  us, 
from  the  sun  and  stars,  across  the  void  of  interplanetary  space, 
necessitate  the  assumption  of  the  ether.     Many  kindred  phe- 
nomena like  that  of  the  transparency  of  vacuum  tubes,  taken 
together  with  the  fact  that  the  physical  properties  of  ordinary 
matter  do  not  enable  us  to  account  for  the  enormous  rapidity 
with  which  light  is  transmitted,  afford    additional    support    to 
the   assumption.      It   has    been   already  shown  in  the  second 
volume  of  this  treatise,  that  the  existence  of  a  similar  medium 
is  necessary  to  the  explanation  of  electro-magnetic  phenomena. 

It  is  universally  considered  that  the  electro-magnetic  ether 
and  the  luminiferous  ether  are  identical. 

In  the  case  of  sound,  which  cannot  reach  us  across  a  vacuum, 
it  is  certain  that  we  have  to  do  with  a  wave-like  disturbance  of 
the  air  or  of  other  material  media.  In  the  discussion  of  wave 
motion,  reflection,  refraction,  interference,  and  diffraction,  it  will 
be  found  advantageous  to  consider  light  and  sound  together ;  in 
other  portions  of  the  volume  they  will  be  treated  separately. 


4  ELEMENTS   OF   PHYSICS. 

612.  The  velocity  of  light  and  the  velocity  of  sound.  —  It  is 

a  familiar  fact  that  sound  requires  a  perceptible  time  to  reach 
the  ear  from  a  sounding  body.  A  Danish  astronomer,  Roemer 
(1675),  was  the  first  to  show  that  the  same  is  true  for  light. 

613.  Methods  of  measuring  the  velocity  of  sound.*  —  The  first 
attempt  to  measure  accurately  the  velocity  of  sound  was  made 
by  a  committee  of  members  of  the  French  Academy  of  Sciences 
in    1738.      The  observers  were  placed  at   night   at  the  Paris 
Observatory,  and  at  three  stations  visible  from  that  point  in 
the  surrounding  country.     Every  ten  minutes  a  cannon  was 
fired  from  one  of  these  stations.     At  the  others  observations 
were  made  of  the  time  which  elapsed  between  the  flash  and  the 
sound  of  the  cannon.     Since  the  distance  between  various  sta- 
tions was  known,  the  velocity  of  sound  could  be  computed.     In 
1822  this  experiment  was  repeated  at  Paris  in  a  slightly  modi- 
fied form ;  two  stations  were  selected,  and  cannon  were  fired 
from  these  alternately  at  intervals  of  ten  minutes.     In  this  way 
the  influence  of  the  wind  was  eliminated.     The  distance  between 
the  two  stations  was  18,622.27  meters.     The  mean  length  of 
time  required  in  one  direction  was  54.84  seconds,  and  in  the 
other  direction  54.43  seconds. 

The  value  of  the  velocity  of  sound  reduced  to  a  temperature 
of  zero  C.  of  the  air  was  found  to  be  331.2  meters  per  second. 
Similar  experiments,  made  near  Amsterdam,  gave  a  velocity  of 

332.26 meters. 

sec. 
As  will  be  shown  in  Chapter  II.,  the  velocity  of  sound  in  a 

given  gas  varies  only  with  the  temperature  of  the- gas,  and  is 
the  same  for -all  pressures  (and  densities)  at  the  same  tem- 
perature. 

Experiments  upon  the  velocity  of  sound  at  low  temperatures, 
made  during  a  winter  spent  in  the  Arctic  regions  by  Lieutenant 
Greeley,  gave  the  following  values  : 

*  For  a  more  complete  account  of  the  researches  described  in  this  article,  see 
Wullner,  Experimentalphysik  (5th  ed.,  Vol.  I,  p.  928). 


LIGHT   AND    SOUND   DEFINED. 


TEMPERATURES. 

VELOCITY. 

-  10.9° 

326.1  meterS 

sec. 

-25-7 

3I7-I 

-37-8 

309.7 

-45.6  • 

305.6 

To  establish  the  fact  that  the  velocity  of  sound  is  independent 
of  the  density  of  the  air,  observations  were  made  by  Bravais  and 
Martins  between  stations  at  the  top  of  the  Faulhorn,  a  mountain 
in  Switzerland,  and  upon  the  shore  of  Lake  Brienz  at  the  foot 
of  that  mountain.  Shots  were  exchanged  between  these  sta- 
tions, the  difference  in  height  of  which  was  2079  meters,  and 
it  was  found  that  the  velocity  of  sound  was  the  same  as  those 
obtained  in  experiments  near  the  level  of  the  sea.  The  average 
velocity  (reduced  to  o°)  was  found  to  be  332.37  meters  per 
second. 

In  all  these  experiments  it  was  assumed  that  the  length  of 
time  required  to  become  aware  of  the  flash  and  of  the  report 
following  it  would  be  the  same ;  while  the  velocity  of  light  is  so 
great,  as  compared  with  that  of  sound,  that  the  time  required  for 
the  light  signal  to  traverse  the  measured  path  is  entirely  negli- 
gible. Now  the  time  required  for  an  observer  to  become  cog- 
nizant of  the  impression  upon  his  optic  or  aural  nerves  is  very 
considerable.  Since,  in  the  case  of  the  flash,  the  observer 
would  be  taken  by  surprise,  and  the  flash  would  act  as  a 
warning  so  that  he  would  be  ready  for  the  reception  of  the 
sound  which  follows  it,  it  was  thought  that  these  methods 
were  open  to  criticism.  The  French  observer,  Regnault, 
therefore  performed  experiments  upon  the  velocity  of  sound 
in  which  the  discharge  was  automatically  recorded  by  causing 
a  bullet  to  cut  a  wire  stretched  across  the  muzzle  of  a  pistol, 
and  in  which  the  arrival  of  the  sound  wave  at  the  receiving 
station  was  likewise  recorded  by  means  of  an  electrical  device. 
The  latter  arrangement,  of  which  a  diagram  is  given  in  Fig. 


ELEMENTS   OF   PHYSICS. 


372,  consisted  of  a  very  sensitive  diaphragm  carrying  a  metal 
disk.  A  screw  P  was  turned  until  its  point  was  almost  in 
contact  with  the  disk.  The  impulse  of  the  sound  wave  was 
sufficient  to  bring  the  two  into  contact,  thus  completing  an 

electric  circuit.  The  records  were 
made  upon  a  chronograph  sheet.  The 
results  obtained  by  this  method  gave 
a  slightly  smaller  value  A  30. 7  meters) 

for  the  velocity  of  sound  than  those 
obtained  in  the  experiments  already 
described.  Taking  all  the  available 
data  together,  the  most  probable  value 
for  the  velocity  of  sound  at  o°  is  found 
to  be  331.76  meters  per  second. 


Fig.  372. 


In  addition  to  the  direct  methods  just  described,  there  are 
various  indirect  methods  of  measuring  velocity  of  sound.  To 
these  reference  will  be  made  in  subsequent  chapters.  These 
methods  are  especially  valuable  in  the  determination  of  the 
velocity  of  sound  in  other  gases  than  air,  and  in  solids  and 
liquids  where  it  is  not  practicable  to  employ  a  path  thousands 
or  even  hundreds  of  meters  in  length.  The  following  table 
gives  the  velocity  of  sound  in  various  substances  in  terms  of 
the  velocity  of  sound  in  air : 


SUBSTANCES. 

VELOCITY  OF  SOUND  COMPARED  WITH 

THAT   IN    AlR. 

Lead 

4-257 

Gold 

6.424 

Silver 

8.057 

Copper 

11.167 

Platinum 

8.467 

Iron 

15.108 

Glass 

15.29 

Water 

4-3 

LIGHT   AND   SOUND   DEFINED.  7 

614.  Methods  of  measuring  the  velocity  of  light.*  —  The  ob- 
servations of  Roemer,  upon  which  the  first  ideas  concerning 
the  velocity  of  light  were  based,  were  of  an  astronomical  char- 
acter. He  found  that  the  observed  time  of  the  revolution  of 
the  satellites  of  Jupiter  varied  according  to  the  position  of  the 
earth  in  its  orbit.  When  the  earth  was  so  situated  as  to  be 
moving  away  from  Jupiter,  the  periodic  time  of  the  satellite 
appeared  greater  than  when  the  earth  was  moving  toward  the 
latter  planet. 

It  was  not  until  the 
present  century  that  at- 
tempts were  made  to 

measure   the  velocity   of  M| 

light   by  direct  methods. 
The  essential  features  of 

Fig.  373. 

the  first  of  these  methods, 

that  employed  by  Fizeau  and  later  by  Cornu,  are  indicated  in 

Fig.  373- 

Light  from  a  brilliant  source,  S,  reflected  from  the  face  of 
an  unsilvered  glass,  is  sent  between  the  teeth  of  a  cogwheel  to 
a  mirror  M,  situated  at  a  great  distance ;  in  the  case  of  Cornu's 
measurements  at  a  distance  of  23  kilometers.  From  this  mirror 
the  light  is  reflected  back  upon  its  path  so  that  the  ray  passes 
through  between  the  same  pair  of  teeth.  If,  now,  light  has  a 
finite  velocity,  and  if  the  distance  between  the  toothed  wheel 
and  the  mirror  is  great,  it  will  be  found  possible  to  drive  the 
wheel  with  sufficient  rapidity  so  that  in  the  interval  during 
which  the  light  is  traveling  from  the  wheel  to  the  mirror 
and  back  again,  the  opening  through  which  it  passed  will  have 
been  supplanted  by  the  next  following  tooth  of  the  wheel.  An 
observer  at  A,  looking  through  the  opening  in  the  wheel,  would 
then  no  longer  be  able  to  see  the  returning  ray,  because  the 
light  passing  between  each  pair  of  teeth  of  the  wheel  to  the 

*  For  a  fuller  account  of  researches  upon  the  velocity  of  light,  see  Preston's  Theory 
of  Light,  Chapter  XIX. 


8  ELEMENTS   OF   PHYSICS. 

mirror  would,  upon  its  return,  be  intercepted  by  the  next  fol- 
lowing tooth.  By  measuring  the  velocity  of  the  wheel  and  the 
distance  between  the -mirror  and  the  wheel,  the  velocity  of  light 
may  be  computed.  The  result  obtained  by  Cornu,  who  has 
made  the  most  trustworthy  experiments  by  this  method,  was 
300,400,000  meters  per  second. 

The  other  method  upon  which  our  knowledge  of  the  velocity 
of  light  depends  is  known  as  Foucault's  method.  In  this 
method  the  ray  of  light  is  reflected  from  a  rapidly  revolving 

mirror  R  (Fig.  374),  to  a  distant 
fixed  mirror  M\  thence  back  to 
the  revolving  mirror.  If  the 
distance  between  these  mirrors 
be  very  great  and  the  velocity 
of  the  mirror  be  high,  it  is  found 

INCIDENT  RAY  ^ that  the  path  of   the  returning 

"""«r~u^7N-— i ^      raY>  represented  by  a  dotted  line 

Fi    374  in  the  diagram,   deviates  meas- 

urably from  that  of  the  incident 

ray.  A  measurement  of  the  angle  between  these  rays,  of 
the  distance  between  R  and  M,  and  of  the  angular  velocity 
of  R,  makes  it  possible  to  compute  the  velocity  of  light.  The 
result  obtained  by  Foucault  was  298,000,000  meters  per  second. 
Foucault's  experiments  have  been  repeated  by  Michelson  and 
by  Newcomb  under  conditions  which  ensured  greater  accuracy 
than  did  the  experiments  of  the  originator  of  the  method. 
Michelson  obtained  as  the  velocity  of  light,  299,853,000  meters 
per  second.  Newcomb,  at  Washington,  found  299,860,000 
meters  per  second. 


/ 


CHAPTER   II. 
WAVES. 

615.  Nature    of    waves.  —  When    a    portion    of    an    elastic 
medium  is  suddenly  distorted,  and  released,  a  wave  of  distor- 
tion passes  out  in  all  directions  from  that  portion  at  a  definite 
velocity.     A  distant  portion  of  the  medium  is  quiescent  until 
this  wave  reaches  it.     It  is  distorted,  and  thrown  into  commo- 
tion, as  the  wave  passes,  after  which  it  again  becomes  quiet. 
The  movement  along  a  stretched  wire,  of  a  wave  produced  by 
a  blow,   and  waves    upon   the  surface   of   water,   are   familiar 
examples. 

616.  Longitudinal  and  transverse  waves.  —  Fluids  can  trans- 
mit only  those  waves  in  which,  as  the  wave  passes,  the  particles 
of  the  medium  move  to  and  fro  in  the  direction  of  progression 
of  the  wave.     Such  waves  are  called  longitudinal  waves.     Thus 
waves  in  the  air,  sound  waves,  are  longitudinal.     On  the  other 
hand,  solids  can  transmit  longitudinal  waves,  and  also  waves  in 
which  the  particles  of  the  medium  move  to  and  fro  in  a  direc- 
tion at  right  angles  to  the  direction  of  progression  of  the  wave. 
Such  waves  are  called  transverse  waves.     Longitudinal  and  trans- 
verse waves  have  different  velocities  of  progression  in  the  same 
medium ;  and,  therefore,  if  a  center  of  disturbance  sends  out 
waves  of   both   kinds,  one   kind   will   outstrip   the  other   and 
become  isolated  from  it.     A  familiar  example  of  this  is  afforded 
by  the  waves  which  pass  through  a  long  and  tightly  stretched 
wire  which  is  struck  sharply  with  a  hammer.     A  person  at  the 
other  end  of  the  wire  will  hear  a  sharp  click  when  the  longitu- 
dinal wave  reaches  him,  and  another  when  the  transverse  wave 

9 


I0  ELEMENTS   OF   PHYSICS. 

reaches  him.  The  air  near  the  hammer  will  also  be  disturbed, 
and  an  air  wave  will  reach  the  person,  giving  a  third  click 
between  the  other  two. 

Light  waves,  which  belong  to  the  same  general  class  as  the 
electric  waves  described  in  Arts.  501,  590,  and  603  (Vol.  II.), 
are  known  to  be  transverse  from  the  phenomena  of  polarization. 

617.   Equations  of  wave  motion. 

(a)  Transverse  waves.  —  Let  the  rectangle  in  Fig.  375  represent  a  portion 
of  thickness  A#  of  an  elastic  medium.  At  a  given  instant  let  it  be  displaced 
and  distorted,  as  shown  by  the  dotted  rhombus  A,  by  a  passing  transverse 

wave.  Let  Y  be  the  upward  displace- 
ment of  the  left  face,  and  Y+  AKthe 
upward  displacement  of  the  right  face 
of  A.  Then  the  distortion  of  A  is  a 

A  y 

shearing  strain,  S,  equal  to  -  ,  and, 

at  the  limit,  the  shearing  strain  at  the 
left  face  of  A  is  : 

c     dY 

Fig.  375.  S  =  -.  (0 

This  shearing  strain  is  accompanied  by  a  shearing  stress  />,  such  that 

P=nS,  (ii) 

where  n  is  the  slide  modulus  of  the  substance.  Let  6"  +  A6*  be  the  shearing 
strain  at  the  right  face  of  A.  Then  P  +  A/*  =  n(S  +  A^)  is  the  shearing 
stress  on  that  face.  Let  a  be  the  area  of  the  right  and  left  faces  of  A.  Then 
a  •  A.r  is  the  volume  of  A,  and  ap  •  A^r  is  its  mass  ;  p  being  the  density  of  the 
substance.  The  force  pulling  downwards  on  the  left  face  of  A  is  Pa,  and  the 
force  pulling  upwards  on  the  right  face  is  (P  +  &P)a.  The  difference  a  •  A/* 
is  an  unbalanced  force  which  must  be  equal  to  the  product  of  the  mass  of  A 

into  its  acceleration  —  -.     Therefore  ap  •  A.r  —  -.=  a  -  A/>;  or 
at1  at2- 

Substituting  in  this  equation  the  value  of  P  from  (ii),  and  then  the  value 
of  S  from  (i),  we  have 


(ti)    Longitudinal  waves.  —  Let  the   heavy  line   rectangle    in    Fig.    376 
represent  a  portion  of  an  elastic  medium  which,  at  a  given  instant,  is  dis- 


WAVES. 


placed  and  distorted,  as  shown  by  the   dotted  rectangle  A,  by  a  passing 
longitudinal  wave.     Let  X  be  the  displacement  of  the  left  face  of  A,  and 


AiC 



X—axis 

A 

Fig.  376. 

X  +  AA"  the  displacement  of  the  right  face  of  A.     Then,  in  a  manner  pre- 
cisely similar  to  the  above,  it  may  be  shown  that 

d*X  =  V 
dP       p 


(v) 


in  which  V  is  the  particular  elastic  modulus  which,  multiplied  by  the  strain 
— — ,  gives  the  corresponding  stress.  For  liquids  and  gases  V  is  the  bulk 
modulus.* 

Solution  of  the  equation  of  wave  motion.  —  Equation  (v)  may  be  written 


d*X 


in  which 


(vi) 
(vii) 


The  solution  of  equation  (vi)  is 

X  =f(x  +  O  +  F(x  -  vf),  (viii) 

in  which /"and  /''signify  any  functions  whatever.     F(x  —vf)  is  a  wave  travel- 
ing in  the  positive  direction  along  the  axis  of  x,  and  f(x  +  vf)  is  a  wave 

traveling  in  the  other  direction.     The  velocity  of  progression  is  v(  =-y— )• 

The  modulus  V  is  the  isentropic  bulk  modulus.     (See  Art.  261,  Vol.  I.) 

Precisely  similar  results  concerning  transverse  waves  may  be  obtained  from 
equation  (iv),  in  which  case  the  velocity  of  progression  is 

(ix) 


*  The  modulus  V  has  the  following  significance  for  a  solid :  Consider  a  strain 
having  but  one  stretch  a.  Associated  with  this  strain  is  a  stress  of  which  all  three 
pulls  are  finite.  The  pull  P,  in  the  direction  of  a,  is  taken  as  the  measure  of  the 

stress,  and  we  have  P  —  Va..     It  is  this  stretch  a  which  is  represented  above  bv 

'    AJC 


I2  ELEMENTS   OF   PHYSICS. 

The  value  of  n  (slide  modulus)  is  zero  for  liquids  and  gases,  whence  it 
follows  that  transverse  waves  are  not  possible  in  liquids  and  gases.  In 
solids,  V  and  n  are,  in  general,  different  in  value,  so  that  longitudinal  and 
transverse  waves  usually  have  different  velocities  in  the  same  medium. 

In  a  gas,  say  air,  the  isentropic  bulk  modulus  V  is  equal  to  kp  (by  Art.  262, 
Vol.  I.),  where  k  is  the  ratio  of  the  two  specific  heats  of  the  gas,  and  p  is  the 
pressure.  Therefore,  writing  kp  for  V  in  equation  (vii),  we  have 


VI 


=  >/T 

in  which  v  is  the  velocity  of  longitudinal  waves  (sound)  in  air,  k  =  1.41,  p  is 
the  pressure,  and  p  is  the  density  of  the  air. 

P 

The  ratio  —  is,  by  Gay  Lussac's  Law  (Art.  249,  Vol.  I.),  proportional  to 

the  absolute  temperature.  Therefore  the  velocity  of  sound  in  air  is  propor- 
tional to  the  square  root  of  the  absolute  temperature ;  that  is : 

vt:v0::  ^273  +  /  :  A/273, 

/273  -1-  / 
or  vt  =  v0  \-27y->  (319) 

in  which  vt  is  the  velocity  of  sound  in  air  at  t°  C,  and  z>0(  =  331.76  -  —  J 
is  the  velocity  at  o°  C. 

618.  Waves  from  periodic  disturbances ;    wave  trains.  —  A 

periodic  disturbance  is  one  which  is  repeated,  in  every  detail,  in 
equal  intervals  of  time.  The  time  interval  r  during  which  one 
repetition  of  the  disturbance  takes  place  is  called  the  period  of 
the  disturbance,  and  the  number  of  repetitions  per  second  is 
called  the  frequency. 

A  periodic  disturbance  sends  out  what  is  called  a  train  of 
waves,  each  one  of  which  is  exactly  like  its  forerunner.  The 
distance  X  between  similar  parts  of  the  adjacent  waves  of  a 
train  is  called  the  wave  length ;  it  is  the  distance  traveled  by 
the  waves  during  the  period  r.  If  the  velocity  of  the  waves 
be  v,  this  distance  is  in,  so  that 

X  =  VT.  (320) 

619.  Graphic  representation  of  wave  trains.  —  Consider  a  wave 
train  traveling  in  the  direction  of  the  line  AB  (Fig.  377).     At 


WAVES. 


each  point  of  AB,  erect  a  perpendicular  (as  a,  b)  whose  length 
is  proportional  to  the  actual  displacement  of  the  medium  at  that 
point  at  a  given  instant.  The  curved  line  ccc,  so  constructed, 
represents  the  wave  train  graphically;  and,  if  this  curve  be 
imagined  to  move  along  at  the  velocity  of  the  wave  train,  the 


Fig.  377. 

actual  motion  of  the  various  parts  of  the  medium  is  clearly  set 
forth.  If  the  train  is  one  of  transverse  waves,  then  the  points 
of  the  curve  give  the  actual  positions  of  the  particles  of  the 
medium  which,  when  the  medium  is  at  rest,  lie  along  the 
line  AB. 

620.   Amplitude ;   phase  ;   energy  stream. 

(a)  Amplitude  of  a  wave  train.  —  The  maximum  displacement 
b  (Fig.   377)  in  a  wave  train  is  called   the   amplitude  of  the 
train. 

(b)  Opposition  in  phase.  — Two  points  in  a  wave  train  at  which 
the  displacements  are  equal  and  opposite  are  said  to  be  opposite 
in  phase.      The  terms  crest  and  hollow,  as  applied  to  water 
waves,  will  be  used  to  signify  those  portions  of  any  wave  train 
where  the  displacements  have  the  greatest  positive,  and  greatest 
negative,  values  respectively. 

(c)  Energy  stream  in  a  wave  train.  —  It  can  be  shown  that, 
for  a  given  medium,  the  energy  per  second  streaming  across  a 
unit  area,  which  is  perpendicular  to  the  direction  of  progression 
of  a  wave  train,  is  proportional  to  the  product  of  the  square  of 
the  amplitude  divided  by  the  square  of  the  wave  length.     This 
energy  stream  is  the  physical   measure  of  the  intensity  of  a 
wave  train.     (Compare  Arts.  598  and  603,  Vol.  II.) 


I4  ELEMENTS   OF   PHYSICS. 

621.  The  principle  of  superposition.  —  It  is  a  familiar  fact  that 
a  number  of  objects  remain  distinctly  visible  to  a  number  of 
observers  when  the  light,  in  passing  from  the  various  objects  to 
the  various  observers,  has  to  cross  the  same  region  at  the  same 
time.     It  is  also  true  that,  although  a  combination  of  sounds  is 
more  or  less  distracting  to  the  attention,  one  sound  does  not 
sensibly  alter  the  character  of  another  which  accompanies  it. 
A  number  of  waves  can  therefore  traverse  the  same  region 
simultaneously,   each   one  independently  of   the  others.      The 
actual  displacement   of  a  particle  of  the  medium  at  a  given 
instant  is  the  vector  sum  of  the  displacements  at  that  instant 
dtie  to  the  separate  waves.     Waves  on  the  surface  of  water  are, 
in  the  same  way,  independent  of  one  another.      An  observer 
overlooking   the   sea   can   trace   simultaneously  the   incoming 
ocean  swell,  the  smaller  waves  caused  by  passing  boats,  the 
waves  due  to  local  wind,  and  the  waves  reflected   from  the 
shore. 

622.  Stationary  wave   trains.  —  Consider  two   similar  wave 
trains  A  and  B,  Fig.  378  (drawn  one  above  the  other  to  avoid 
confusion),  moving  in  opposite  directions,  as  indicated  by  the 
arrows.     By  the  principle  of  superposition,  the  actual  displace- 
ment at  each  point  is  equal  to  the  sum  of  the  displacements  at 
that  point  due  to  each  wave  train,  and  the  actual  velocity  of 
each   particle   is   equal   to   the   sum  of   the  velocities  of  that 
particle  due  to  each  wave  train.    Therefore  the  medium  remains 
stationary  along  the  lines  // ;  for  the  ordinates  O,  which  come 
up  to  the  line  pp  as  the  upper  train  moves  to  the  right,  are 
at  each  instant  equal  and  opposite  to  the  ordinates  O'  which 
come  up  to  the  line  pp  as  the  lower  train  moves  to  the  left. 

The  portions  of  the  medium  between  the  lines  //  move  up 
and  down  (to  right  and  left  in  case  of  a  longitudinal  wave),  for 
the  ordinates  Q  and  Qr,  which  come  into  successive  coincidence 
at  q,  are  of  the  same  sign.  The  stationary  portions  of  the 
medium  are  called  nodes,  and  the  intermediate  vibrating  por- 


WAVES.  j  - 

tions  are  called  vibrating  segments.  The  middle  point  of  a 
vibrating  segment  is  called  an  antinode.  The  resultant  of  the 
two  wave  trains  A  and  B  is  called  a  stationary  train.  '  The  result- 
ant curve  (Fig.  378)  shows  the  character  of  this  stationary  wave 
train.  This  resultant  curve  is  drawn  below  A  and  B  to  avoid 


Q 


P 


;p 


Fig.  378. 

confusion.  The  straight  line  is  the  resultant  of  the  trains  A 
and  B  when  they  are  in  the  positions  shown,  and  the  arrows 
represent  the  velocities  at  each  point.  The  lines  i,  2,  (3),  (4), 
5,  6,  and  (7)  represent  the  successive  stages  of  the  motion  as 
the  trains  A  and  B  move  to  right  and  left. 

The  nodes  of  a  stationary  train  are  evidently  the  places  where 
the  two  trains  A  and  B  are  always  opposite  in  phase,  and  the 
antinodes  are  the  places  where  the  two  trains  A  and  B  are 
always  in  the  same  phase. 


i6 


ELEMENTS   OF   PHYSICS. 


The  portions  of  the  medium  at  the  antinodes  of  a  stationary 
wave  train  have  at  times  considerable  velocity,  but  are  never 
distorted.  "The  portions  at  the  nodes  never  move,  but  are  at  times 
much  distorted.  This,  for  transverse  waves,  is  shown  in  Fig. 
379.  The  shaded  areas  represent  portions  of  the  medium.  AB 
represents  the  state  of  affairs  when  the  velocities  in  the  vibrat- 
ing segments  are  greatest,  as  indicated  by  the  arrows,  and  when 
the  displacements  are  everywhere  zero.  CD  represents  the 
state  of  affairs  one-quarter  of  a  period  later,  when  the  velocities 
are  everywhere  zero  and  the  displacements  are  at  their  greatest. 
The  shaded  area  at  the  node  is  distorted  as  shown. 


a 


Pi 


P  Q  PI 

Fig.  379. 

Stationary  wave  trains  may  result  from  the  superposition  of 
wave  trains  of  any  shape,  provided  only  that  the  advancing  train 
is  exactly  similar  to  the  receding  one  turned  end  for  end  and 
upside  down. 

623.  Stationary  wave  trains  by  reflection.  —  When  a  wave 
train  reaches  the  boundary  of  a  medium,  it  is  reflected.  The 
reflected  train  is  similar  to  the  incident  train,  and  if  the  incident 
train  strikes  the  boundary  normally,  then  a  stationary  train  is 
produced  by  the  superposition  of  the  two. 

If  the  boundary  were  one  which  separated  the  medium  from 
a  void,  then  the  surface  layers  of  the  medium  could  not  be  dis- 


WAVES.  i7 

torted,  since  they  would  be  perfectly  free  to  move  with  the 
contiguous  portions  of  the  medium.  In  such  a  case  there  would 
be  an  antinode  of  the  stationary  train  at  the  boundary.  If  the 
boundary  were  a  rigid  wall,  then  the  medium  contiguous  to  the 
wall  would  not  be  free  to  move,  and  there  would  be  a  node  of 
the  stationary  train  at  the  boundary.  In  all  intermediate  cases 
it  may  be  stated  that  where  the  boundary  separates  the  medium 
from  some  material  less  dense  than  itself,  an  antinode  is  formed 
at  the  boundary,  and  where  the  boundary  separates  the  medium 
from  a  denser  material,  a  node  is  formed. 

The  component  wave  trains  of  a  stationary  train  are  opposite 
in  phase  at  a  node,  and  in  phase  at  an  antinode,  so  that  a  reflected 
wave  train  is  in  phase  with  the  incident  train  at  a  free  boundary, 
and  opposite  in  phase  at  a  rigid  boundary.  Reflection  of  the 
first  kind  is  called  reflection  without  change  of  phase,  and  reflec- 
tion of  the  second  kind  is  called  reflection  with  change  of  phase. 
A  surface  separating  two  media  in  which  the  velocities  of  a  wave 
are  different  reflects  with  change  of  phase  in  the  medium  giving 
the  greater  velocity,  and  without  change  of  phase  in  the  other 
medium. 

Stationary  wave  trains  may  be  easily  produced  by  means  of  a 
stretched  cord  or  wire,  or  with  a  stretched  rubber  tube.  One 
end  of  the  rubber  tube  is  tied  to  a  rigid  support,  and  the  other 
end  is  held  in  the  hand.  A  series  of  periodic  movements  of  the 
hand  generates  a  wave  train  on  the  tube.  This  train  is  reflected 
from  the  rigid  end  with  change  of  phase ;  and  the  tube  is 
broken  up  into  segments  with  intervening  nodes  as  the  reflected 
train  and  advancing  train  come  into  superposition.  The  rigid 
end  of  the  tube  is  a  node. 

A  stationary  wave  train  may  likewise  be  produced  in  the  air 
within  a  long  glass  tube.  In  this  experiment  both  ends  of  the 
tube  are  open.  The  lips,  applied  to  one  end  of  the  tube  as  to 
a  bugle,  are  thrown  into  periodic  motion  producing  a  wave  train, 
which  is  reflected  from  the  open  end  without  change  of  phase. 
A  stationary  wave  train  is  produced  in  the  tube  when  the  re- 


i8 


ELEMENTS   OF   PHYSICS. 


mood 


a/e 


fleeted  train  and  advancing  train  come  into  superposition.  The 
open  end  of  the  tube  is  the  center  (nearly)  of  a  vibrating  seg- 
ment. If  lycopodium  powder  is  strewn  along  the  inside  of  the 

tube,  it  is  swept  into  the  nodal 
points  by  the  violent  to  and  fro 
motion  of  the  air,  giving  a  strik- 
ing indication  of  the  existence 
of  the  stationary  train.  (Com- 
pare Art.  790.) 


far 


624.  Simple  and  compound 
wave  trains.  —  When  the  curve 
f*u  which  represents  a  wave  train 
graphically  is  a  curve  of  sines, 
-;<€  the  wave  train  is  said  to  be  sim- 
ple, otherwise  the  train  is  said 
to  be  compound.  The  curves  of 
Fig.  380*  represent  the  wave  trains  which  issue  from  the 
mouth  when  the  indicated  vowel  sounds  are  produced  by  a 
baritone  voice.  The  curve  No.  i  is  a  simple  train.  The  others 
are  compound. 


Fig.  380. 


625.  Fourier's  theorem. f —  A  compound  wave  train  is  the 
superposition  of  a  series  of  simple  wave  trains,  of  which  the 
respective  wave  lengths  are  i,  J,  J-,  \,  -J,  etc.,  of  the  wave  length 
of  the  given  compound  train,  and  of  which  the  respective  am- 
plitudes are  determinate. 

Example. — The  heavy  line  AB  (Fig.  381)  represents  one 
wave  of  a  wave  train,  which  is  compounded  of  the  simple  wave 
train  represented  by  the  dotted  line  AB  and  another  simple 
wave  train  CD,  of  half  the  wave  length.  The  simple  trains  of 


*  These  curves,  by  Mr.  L.  B.  Spinney,  are  copies  of  photographic  tracings. 
t  See  Fourier's  Series  and  Sperical  Harmonics,  W.   E.  Byerly,  pp.  30-38,  for  a 
discussion  of  Fourier's  Theorem. 


WAVES.  I9 

which  the  curves  in  Fig.  380  are  built  up  have  been  studied  by 
Helmholtz  and  others. 


x'B 


Fig.  381. 

626.  Wave  front.  —  Consider  a  region,  AB  (Fig.  382),  which 
is  disturbed  by  a  wave  coming  from  a  distant  center  of  disturb- 
ance C.     If  the  medium  is  isotropic,  then  all  parts  of  any  small 
plane  layer  AB  of  the  medium  perpendicular  to  r  will  be  similarly 
distorted,  and  the  layer  will  move  A 

as  a  whole  up  and  down  or  to  Q^ v_ — 

and  fro  as  the  wave  passes.     A 

Fig.  382.  B 

surface    so    drawn    as    to    pass 

through  those  parts  of  a  wave  where  the  distortion  is  everywhere 
the  same,  or  those  parts  where  the  displacement  is  everywhere  the 
same,  is  called  a  wave  front. 

The  direction  of  progression  of  a  wave  in  an  isotropic  medium 
is  at  right  angles  to  its  front. 

When  a  wave  has  come  from  .a  large  number  of  centers  of 
disturbance  near  at  hand,  it  has  no  definite  front.  Such  a  wave 
may,  however,  be  looked  upon  as  a  superposition  of  elementary 
waves  coming  each  from  a  center  of  disturbance,  and  these  ele- 
mentary waves  have  definite  fronts.  A  wave  having  a  definite 
front  may  lose  this  character  upon  striking  an  obstacle. 

627.  Huygens'    principle.  —  Let  AB  (Fig.  383)   be   the   in- 
stantaneous position  of  a  wave  which  has  come  from  a  disturb- 


20  ELEMENTS   OF   PHYSICS. 

ance  at  C.     The  disturbance  produced  later  at  /  as  the  wave 
passes  that  point  is,  of  course,  to  be  thought  of  as  having  come 

originally  from  C ;  it  may,  however, 
be  considered  to  have  come  from 
the  disturbance  which  constitutes 
x  the  wave  AB.  In  this  latter  case 
each  point  of  the  wave  AB  is  to  be 
considered  as  a  center  of  disturb- 
ance from  which  a  spherical  wave 
emanates.  The  waves  which  thus 
emanate  from  each  point  of  a  prim- 
ary wave  are  called  secondary  waves  or  wavelets.  The  actual 
disturbance  produced  at  /  is  the  superposition  of  the  effects  of 
all  these  wavelets. 

628.    Huygens'  construction  for  wave  front.  —  Let  AB  (Fig. 
384)  be  the  front  of  a  wave  advancing  toward  A' BJ .     Let  it  be 
required  to  find  the  wave  front  after  a  time  has  elapsed,  during 
which  the  wave  has  traveled  a  distance  r.      Describe  circles 
(spheres)  of  radius  r  from  each  point  of  the 
wave  front  AB.     The  envelope  A'B'  of  these 
circles  (spheres)  is  the  required  wave  front. 
These  spheres  are  the  secondary  wavelets 
described  in  the  previous  article. 

Each  of  the  secondary  wavelets  ema- 
nates from  an  infinitesimal  portion  of  AB, 
\  ,  and  represents  an  infinitesimal  disturbance. 
The  number  of  these  secondary  wavelets 
which  work  together  (i.e.  in  the  same 
phase)  to  produce  disturbance  in  a  portion 
of  their  enveloping  surface  A'B'  is  infinitely  greater  than  the 
number  which  work  together  to  produce  a  disturbance  else- 
where. Therefore,  the  disturbance  along  A'B'  is  infinitely 
greater  than  elsewhere.  This  is  equivalent  to  saying  that  the 
disturbance  elsewhere  is  zero. 


WAVES. 


21 


629.    Half-period  zones.  —  Consider  a  simple  train  of   plane 
waves,  of  very  short  wave  length  X,  approaching  the  point  O 
(Fig.  385),  as  indicated  by  the  arrow,  the  wave  fronts   being 
parallel  to  AB.     This  line  AB  repre- 
sents a  fixed  plane  perpendicular  to 
the  paper.     From  O  as  a  center,  im- 
agine spheres  to  be  described  of  which 
the  first  has  a  radius  b,  equal  to  OP ; 

the  next  has  a  radius  b  +  -;  the  next 

2 

radius    #  +  X:    the    next   a   radius 


£  +  ^,  etc. 

2 

divided  into  zones. 


From  the  figure  we  have  rf2  4- 


Fig.  385. 


a   radius    0-t-A,;    tne    next   a 

The  plane  AB  is  thus 

The  first  of  these 
zones  is  a  circle  of  radius  / ;  the  second  zone  is  a  circular  ring 
of  inside  radius  r'  and  outside  radius  ru ;  the  third  zone  is  a 
circular  ring  of  inside  radius  r"  and  outside  radius  r'",  etc. 

/  ^\2 

=  f  b  4-  —  j  ;  whence,  since  X2 


is  negligible  in  comparison  with  b\  we  have 


similarly, 


r"    =V2^X, 


(321) 


etc., 


etc. 


The  nth  zone  is  called  a  zone  of  high  order  when  n  is  a  large 
number.  The  intensity  of  the  disturbance  reaching  O  from 
each  zone  is  slightly  less  than  that  coming  from  the  zone  of 
the  next  higher  order,  but  two  adjacent  zones  of  high  order 
produce  at  O  disturbances  which  are  sensibly  equal  and  opposite 
in  phase,*  so  that  their  effect  is  to  annul  each  other.  There- 

*  The  secondary  wavelets  reaching  O  at  the  same  instant  from  the  nth  and  the 
(»  +  i)th  zones  are  on  the  whole  opposite  in  phase;  for,  the  wavelets  coming  from 
the  (n  -f-  i)th  zone,  having  had  -  farther  to  travel,  must  have  started  half  a  period 
earlier  than  the  wavelets  from  the  nth  zone. 


UNIVERSITY 


22  ELEMENTS   OF   PHYSICS. 

fore  the  actual  disturbance  at  O  comes  from  the  zones  of  low 
order j  and  when  X  is  small  these  are  included  in  a  small  portion 
of  the  plane  AB  surrounding  the  point  P. 

If  a  small  obstacle  is  placed  at  P  so  as  to  cut  off  those  por- 
tions of  the  advancing  waves  from  which  most  of  the  disturb- 
ance at  O  comes,  the  disturbance  at  O  will  cease.  This  is  the 
explanation,  by  means  of  the  wave  theory,  of  the  (sensibly)  recti- 
linear propagation  of  light. 

Examples.  —  (a)  The  value  of  X  for  yellow  light  is  about 
6.io~5  centimeters.  If  b  (Fig.  385)  is  10  centimeters,  the  diam- 
eter of,  say  the  tenth  half-period  zone  is  about  0.16  centimeter, 
which  is  the  order  of  magnitude  of  an  obstacle  required  in  this 
case  to  screen  the  point  O  (Fig.  385). 

(b)  The  value  of  X  for  sound  waves  from  a  shrill  whistle  is, 
say,  9  centimeters.  If  b  (Fig.  385)  is  10,000  centimeters  the 
diameter  of  the  tenth  half -period  zone  is  1880  centimeters,  which 
is  the  order  of  magnitude  of  an  obstacle  required  to  screen  the 
point  O  in  this  case. 

630.  The  ray  of  light.  —  Consider  a  train  of  waves  of  wave 
length  X,  and  of  which  the  fronts  are  parallel  to  AB  (Fig.  386). 
When  X  is  small,  the  disturbance  reaching  O  comes,  sensibly, 

from  the  immediate  neighborhood 
of  P.     This  disturbance  therefore 
travels  along  the  line  PO. 
— > —  Lines  drawn  normal  to  a  wave 

^  ~~  front  of  a  wave  train  along  which 

~~ ~~~ ~~ ~ "7"* "~  °__  the  disturbance  travels  are  called, 

l~H11_1.1_1^r™Z.1^rr in  geometrical  optics  rays.     Rays 

I~1Z.~I_"IZIin~^r_I  drawn  from  every  point  of  a  small 

*  portion  of  a  wave  front  constitute 

B  Fig.  386. 

a  pencil  of  rays. 
Sound  rays.  —  The  wave  length  of  sound  waves  is  ordinarily 
from  5  to  500  centimeters.     Therefore  the  disturbance  reaching 
O  (Fig.  386)  comes,  sensibly,  from  a  portion  of  AB  which  is  so 


WAVES.  23 

large,  compared  with  the  dimensions  of  objects  which  we  en- 
counter in  daily  life,  that  we  do  not  think  of  sound  as  being 
propagated  even  approximately  in  straight  lines.  An  obstacle 
placed  between  P  and  O,  as  has  been  explained  in  Art.  629, 
must  needs  be  very  large  to  screen  O. 


631.  Shadows.  —  When  the  light  from  a  luminous  body  is 
obstructed  by  an  opaque  body,  as,  for  example,  when  the  light 
of  a  lamp  is  obscured  by  the  hand,  the  region  beyond  the  opaque 
body  is  more  or  less  darkened.  This  region  is  called  a  shadow. 

Similarly,  the  sound  of  a  whistle,  for  example,  is  more  or  less 
weakened  in  the  region  behind  a  building.  Such  a  region  is 
called  a  sound  shadow. 

Geometrical  shadows.  —  Consider  a  luminous  point  O  (Fig. 
387).  If  light  were  propagated  in  straight  lines,  the  region 
below  the  line  OB  would  be 


GEOMETRICAL 
SHADOW 


OBSTACLE 


Fig.    387. 


entirely  obscured  by  the  ob-    LUMINOUS 

*  J  POINT 

stacle.  This  sharply  defined 
region  is  called  the  geometri- 
cal shadow.  The  boundary  of 
the  actual  shadow  cast  by  a 
luminous  point  is,  however, 
not  sharp,  because  of  the  bending  of  the  light  into  the  region 
of  the  shadow,  as  described  in  the  chapter  on  Diffraction.  This 
indistinctness  of  boundary  of  a  shadow  is  very  noticeable  in 
sound  shadows  because  of  the  greater  wave  length. 

Umbra  and  penumbra.  — 
Figure  388  represents  the 
shadow  of  the  moon.  The 
region  V,  from  every  part  of 
which  the  sun  would  be  en- 
tirely invisible  to  an  observer, 
is  called  the  umbra.  The 

region   P,  from   every  part   of  which  the   sun   would  be  only 
partially  visible  to  an  observer,  is  called  the  penumbra.      The 


Fig.  388. 


24  ELEMENTS   OF   PHYSICS. 

umbra  and  penumbra  may  be  easily  distinguished  in  the  shadow 
of  the  hand  by  lamplight. 

In  all  ordinary  cases  the  bending  of  light  into  the  region  of 
a  shadow  is  masked  by  the  penumbra,  and  only  becomes  notice- 
able when  light  is  employed,  the  source  of  which  is  very  small 
(a  point). 

632.  Homocentric  pencil.  —  When  a  small  portion  of  a  wave 
front  is  a  sector  of  a  spherical  surface,  the  rays  from  the  por- 
tion pass  through  the  center  of  the  sphere,  and  the  pencil  of 
rays  is  said  to  be  homocentric.  The  center  of  the  sphere  is 


Fig.  389.  Fig.  390. 

called  the  focal  point  of  the  pencil.  When  the  center  of  the 
spherical  wave  is  behind  the  wave,  as  shown  in  Fig.  389,  the 
pencil  is  said  to  be  divergent.  When  the  center  of  the  spheri- 
cal wave  is  in  front  of  the  wave,  as  shown  in  Fig.  390,  the 

pencil  is  said  to  be  convergent 

A  -r^Tl — I — I — T->  ^  Penc^  °f  parallel  rays  is  a 
homocentric  pencil  with  its 
center  at  infinity. 

633.    Astigmatic     pencil.  — 

Consider  an  arc  of  a  circle 
A  A  (Fig.  391)  with  its  center 
at  C.  Imagine  this  arc  to  be 
rotated  about  the  line  DD  as 

Fig.  391.  Fig.  392.  an      aXlS'        The      *™     AA     WU1 

describe   a   surface,   of   which 

the  principal  sections  are  shown  in  Figs.  391  and  392.  It  is 
shown  in  treatises  on  geometry  that  a  small  portion  of  any 


WAVES.  25 

surface  whatever  is  either  a  portion  of  a  sphere  (the  osculating 
sphere)  or  it  is  of  the  shape  of  A  A  (Figs.  391  and  392.) 

If  a  small  portion  of  a  wave  front  is  of  the  shape  of  AA 
(Figs.  391  and  392),  all  the  rays  drawn  from  it  will  pass  through 
the  line  DD  and  through  the  line  CC.  Such  a  pencil  of  rays 
is  called  an  astigmatic  pencil.  The  central  ray  of  the  pencil  is 
called  the  axis  of  the  pencil.  The  lines  CC  and  DD  are  called 
the/<?oz/  lines  of  the  pencil. 


CHAPTER   III. 
REFLECTION   AND   REFRACTION. 

634.  Regular  and  diffuse  reflection ;  refraction.  —  When  light 
falls  upon  the  surface  of  a  body,  it  is  in  part  reflected.     When 
reflected  light  enters  the  eye  from  a  polished  surface,  we  see 
the  objects  from  which  the  light  has  originally  come.     On  the 
other  hand,  we  see  only  the  reflecting  body  if  its  surface  is 
rough.      Reflection   from   polished   surfaces   is   called   regular 
reflection.     This  kind  of  reflection  is  discussed  in  the  following 
articles.      Reflection    from    rough    surfaces    is    called    diffuse 
reflection. 

When  light  enters  one  medium  from  another  across  a 
polished  surface,  as  for  example  when  light  enters  glass  or  water 
from  air,  the  direction  of  progression  of  the  light  is,  in  general, 
altered.  This  phenomenon  is  called  refraction. 

635.  The  application  of  Huygens'  principle  to  reflection  and 
refraction.  —  Let  AB  (Fig.  393)   be  a  surface  separating  any 
two  media,  say  air  and  glass.      Let  v  be  the  velocity  of  light 
waves  in  the  glass  and  JJLV  their  velocity  in   air;    fj,  being  a 
constant.     Consider  a  wave  approaching  AB  from  (7,  and  let 
WW  be  the  position  this  wave  would  reach  at  a  given  instant 
were  it  not  for  the  discontinuity  at  AB. 

Each  point  of  AB  may  be  thought  of  as  sending  out  a  secondary 
wavelet  at  the  instant  at  which  the  acttial  wave  reaches  that 
point. 

At  the  given  instant  the  wavelet  from  each  point  will  be 
a  circle  (sphere)  tangent  to  WW —  at  least  that  portion  of  the 

26 


REFLECTION  AND  REFRACTION.  27 

wavelet  which  does  not  cross  AB  will  be  such  a  sphere.  Let 
the  radius  of  the  wavelet  from  /  be  r,  as  shown  in  Fig.  393  (a). 
The  portion  of  the  wavelet  from  /  which  does  cross  AB  will 


GLASS 


Fig.  393  (a). 


Fig.  393  (b). 


Fig.  393  (c). 


be  a  sphere*  of  radius  rl  =  — ;  for  the  wave  will  have  traveled 

i  * 

only  -  as  far  in  the  glass  as  it  would  have  traveled  in  air 

in  the  same  time. 

Figure  393  (b)  shows  Huygens'  construction  for  the  reflected 
wave.  In  this  case  the  wavelet  from  each  point  of  AB  is  a 
sphere  tangent  to  WW.  The  envelope  W  W  is  the  reflected 
wave.  Figure  393  (c)  shows  Huygens'  construction  for  the 
refracted  wave.  In  this  case  the  wavelet  from  each  point  is 

a  sphere  of  which  the  radius  is  -  as  great  as  it  would  be  to 

make  it  tangent  to  WW.     The  envelope  W  W1  is  the  refracted 
wave. 

636.  Reflection  of  plane  waves  from  a  plane  surface.  —  Con- 
sider a  plane  wave,  WW  (Fig.  394),  approaching  a  plane 
obstacle,  AB.  Let  W1  W  be  the  position  this  wave  would 
have  at  a  given  instant  were  it  not  for  AB.  The  envelope 
W"Wn  is  the  reflected  wave. 

*  In  crystalline  substances  this  portion  of  the  wavelet  is  double.  In  Iceland 
spar  one  portion  is  a  sphere  and  the  other  is  an  ellipsoid. 


28 


ELEMENTS    OF   PHYSICS. 


The  angle  i  and  the  angle  r  in  the  diagram  are  equal.  The 
angle  between  the  incident  ray  and  the  normal  to  AB,  which 
is  equal  to  i,  and  the  angle  between  the  reflected  ray  and  the 

normal  to  AB,  which  is  equal  to  r, 
are   called   respectively  the  angle  of 
incidence  and  the  angle  of  reflection. 
Equality  of  the  angles  of  incidence 


Fig.  395. 


and  of  reflection.  —  The  angle  of  incidence  and  the  angle  of 
reflection,  just  defined,  are  always  equal  whatever  the  shape  of 
the  incident  wave  and  whatever  the  shape  of  the  reflecting 

surface  may  be.  For  let  AB 
(Fig.  395)  be  a  small  portion, 
sensibly  plane,  of  any  reflect- 
ing surface;  and  WW  a.  small 
portion  of  any  incident  wave. 
Then  WW  will  be  reflected 
,B  as  a  plane  wave  from  a  plane 
obstacle. 


637.   Reflection  of  spherical 
waves  from  a  plane  surface.  — 

Let  O  (Fig.  396)  be  a  lumi- 
nous point  from  which  spher- 
ical waves  emanate,  and  AB 
a  plane  reflecting  surface. 
Let  WW  be  the  position  which  would  be  reached  at  a  given 


| 

*, 
0 

Fig.  396. 


REFLECTION  AND  REFRACTION. 


29 


instant  by  a  wave  from  O  were  it  not  for  AB.  The  envelope 
Wr  W  is  the  reflected  wave.  This  reflected  wave  is  evidently 
a  circle  (sphere)  of  which  the  center  is  at  the  point  O' ;  the 
line  OO'  being  perpendicular  to  AB  and  bisected  thereby.  The 
waves  reflected  from  AB  (Fig.  397)  are  portions  of  spheres  with 

their  centers  at  Of. 
0. 

638.  Image  of  an  object  in  a 
plane  mirror.  —  Consider  a  group 
of  luminous  points  O  (Fig.  398). 


Fig.  397. 


Fig.  398. 


The  light  from  these  points,  after  reflection  from  the  plane  sur- 
face AB,  or  from  any  portion  of  it,  appears  to  have  come  from 
a  similar  group  of  points  O1,  as  is  evident  from  Art.  637.  The 
group  of  points  O  is  an  object,  and  the  group  Of  is  its  image  in 
the  mirror  AB. 

639.  Spherical  mirror.  Reflection  of  a  plane  wave.  —  Let 
MM  (Fig.  399)  be  a  spherical  mirror  of  which  the  center  of 
curvature  is  at  R.  Consider  an  incident  plane  wave  coming 
from  the  right,  and  let  the  line  WW  be  the  position  which  this 
wave  would  reach  at  a  given  instant  were  it  not  for  the  mirror. 
The  envelope  W1  W  is  the  reflected  wave.  The  central  por- 
tion of  this  reflected  wave  is  very  nearly  spherical ;  its  center 
of  curvature  is  at  F.  A  continuation  of  this  sphere  is  shown 
at  SS,  which  shows  that  the  reflected  wave  deviates  from  a 
true  spherical  shape.  This  deviation  is  less  than  the  width 


I 


30  ELEMENTS   OF  PHYSICS. 

of  the  line  at  /,  and  it  grows  rapidly  greater  beyond  55.  All 
the  rays  drawn  from  the  reflected  wave  W  Wf  do  not  pass 
through  F.  The  envelope  of  these  rays  is  a  curve,  called  a 
caustic,*  which  has  a  cusp  at  F. 

The  rays  drawn  from  a  small  portion  AA  of  the  reflected 
wave  form  an  astigmatic  pencil,  of  which  one  focal  line  is  at 


w 


Fig.  399. 

C  on  the  caustic  curve  perpendicular  to  the  paper,  and  the 
other  focal  line  is  a  short  portion  DD  of  the' line  VR. 

The  line  VR  is  called  the  axis  of  the  mirror.  The  point  V 
is  called  the  vertex  of  the  mirror. 

A  spherical  mirror  with  its  reflecting  surface  on  its  concave 

*  This  caustic  is  an  epicycloid.     See  Preston's  Light,  p.  90. 


REFLECTION  AND  REFRACTION. 


OF    TTBH 

•DIVERSITY 


side  is  called  a  concave  mirror.     When  the  reflecting  surface 
is  convex,  the  mirror  is  called  a  convex  mirror. 

Plane  waves  (parallel  rays) 'reflected  from  a  parabolic  mirror 
become  spherical  and  are  concentrated  at  its  focus,  and  rays 
diverging  from  its  focus  are  rendered  parallel  after  reflection, 
as  shown  in  Fig.  400.  Such  reflectors  are  used  for  search 
lights  and  for  locomotive  headlights.  They  give  a  beam  of 


Fig.  401. 

light  which  does  not  diverge,  and  which  retains  its  intensity 
for  great  distances. 

An  elliptical  mirror  concentrates  at  one  focus  the  light  which 
comes  from  the  other  focus,  as  shown  in  Fig.  401.  Vaulted 
ceilings  often  approximate  so  nearly  to  an  elliptical  form  that 
a  faint  sound  produced  at  one  point  is  concentrated  at  another 
point  where  it  becomes  distinctly  audible. 

640.  Spherical  mirror  of  small  aperture.  —  A  mirror  has 
small  aperture  when  its  diameter  is  small  compared  with  its 
radius  of  curvature.  Thus  the  central  portion  of  the  mirror 
described  in  Art.  639  would  be  a  mirror  of  small  aperture. 
Spherical  (or  plane)  waves  remain  sensibly  spherical  after  reflec- 
tion from  such  a  mirror,  provided  the  incident  rays  are  not 
greatly  inclined  to  the  axis  of  the  mirror.  Light  coming  from 
a  point  near  the  axis  of  a  mirror  is;  therefore,  after  reflection, 


ELEMENTS   OF   PHYSICS. 


concentrated  at  (or  appears  to  have  come  from)  another  point; 
such  a  pair  of  points  are  called  conjugate  foci.  Plane  waves, 
or  parallel  rays,  are  concentrated  at  a  point  F  (Fig.  399)  called 
the  principal  focus  of  the  mirror.* 

641.  Proposition.  —  The  relation  between  the  radius  of  curva- 
ture R  of  a  mirror,  and  the  distances  a  and  b  of  a  pair  of 
conjugate  foci  from  the  vertex  of  the  mirror  is 


R     a     b 


(322) 


Proof,  f — Let  MM  (Fig.  403)  be  the  mirror;  WW  the  posi- 
tion which  would  be  reached  by  the  incident  wave  at  a  given 
instant  were  it  not  for  the  mirror ;  and  W'  W'  the  reflected 

wave.  Consider  a  small  portion 
of  each,  MM,  WW,  and  W  W, 
of  which  the  common  chord  is  d. 
Let  h  be  versed  sine  OM\  then 


Let  k  be  the  versed  sine  O  W 
of  the  incident  wave,  which  has 
come  from  a  point  distant  a  from 
V\  then  2 

«-£•  (") 


Fig.  403. 


*  Remark.  —  Any  ray  parallel  to  the  axis  of  a  mirror  of  small  aperture  passes 
after  reflection  through  its  principal  focus.  Hereafter  all  mirrors,  unless  otherwise 
specified,  are  understood  to  be  of  small  aperture. 

t  Preliminary  proposition.  —  The  radius  of  curvation,  R  (Fig.  402),  of  a  circular 
arc,  of  which  the  chord  is  d  and  the  versed  sine  is  h,  is 


Fig.  402. 


When  h  is  small  compared  with  d,  this  equation  becomes 


This  equation  (ii)  will  be  used  throughout  the  discussion  of 
reflection  and  refraction. 


REFLECTION  AND  REFRACTION. 


33 


'    Since  WM=MW,  the  versed    sine  OW  of   the  reflected 
wave  is  2  h  —  k,  and  we  have 


in  which  b  is  the  radius  of  curvature  of  the  reflected  wave  or 
the  distance  from  V  at  which  it  will  be  concentrated.  Substi- 
tuting these  values  of  R,  a,  and  b  in  equation  (322),  that  equa- 
tion is  found  to  be  identically  satisfied.  Q.E.D. 

642.    Virtual  and  real  foci.*  —  When  light  from  a  point  is 
actually  concentrated  at   the   conjugate   focus,   the   conjugate 


Fig.  404. 

focus  is  said  to  be  real;  when  after  reflection  from  the  mirror 
the  light  merely  appears  to  have  come  from  the  conjugate 
focus,  the  latter  is  said  to  be 
virtual.  Virtual  foci  are  al- 
ways back  of  the  mirror.  Fig- 
ures 404  and  405  illustrate 
real  and  virtual  foci. 


643.  Principal  focal  length 
of  a  mirror.  —  When  one  of 
a  pair  of  conjugate  foci  is  infinitely  distant  (a,  equation  (322), 
equal  to  infinity),  the  other  is  called  the  principal  focus,  and  its 
listance  from  the  mirror  is  called  the  principal  focal  length, 
p,  of  the  mirror.  Writing  /  for  b  and  infinity  for  a  in  equation 
(322),  we  have  „  T  R 


—  =  -,  or  p  =  — . 
R     p  2 

*  These  definitions  apply  to  the  foci  of  lenses  also. 


(323) 


34 


ELEMENTS   OF   PHYSICS. 


Fig.  406. 


644.  Conjugate  foci  out  of  axis.  —  A  pair  of  conjugate  foci 
which  do  not  lie  on  the  axis  of  a  mirror  lie  on  a  straight  line 
passing  through  the  center  of  curvature  of  the  mirror. 

Proof.  —  Let  MM  (Fig. 
406)  be  a  mirror  of  small 
aperture,  R  its  center  of 
curvature,  and  a  a  lumi- 
nous point.  All  the  rays 
from  a  pass  through  its 
conjugate  b  after  reflection. 
Consider  «the  ray  from  a 
which  passes  through  R. 
This  ray  falls  upon  the  mirror  normally,  is  turned  back  upon 
itself,  and  passes  through  b.  Therefore  b  is  on  the  line 
aR.  Q.E.D. 

Geometrical  construction  for  the  conjugate  of  a  point.  —  Con- 
sider the  ray  aX  from  a  (Fig.  406)  which  is  parallel  to  the  axis 
of  the  mirror.  After  reflection,  this  ray  passes  through  the 
principal  focus  F  (see  Art.  640,  footnote),  and  also  through 
b.  Therefore  the  point  b  is  determined  as  the  point  of  inter- 
section of  the  lines  aR  and  XF. 

645.  Conjugate  planes.  —  Points  (luminous  points)  which  lie 
near  the  axis  of  a  mirror  in  a  plane  perpendicular  to  the  axis 
have   conjugate  points   which   are   similarly  situated   (perhaps 

inverted)  near  the 
axis  in  another  plane 
perpendicular  to  the 
axis,  as  can  be  shown 
by  the  above  geo- 
metrical construction. 
Two  such  planes  are 
called  conjugate  planes. 


Fig.  407. 


646.    Images  formed  by  spherical   mirrors.  —  The   group  of 
points  which  are  conjugate  to  the  respective  points  of  an  object 


REFLECTION  AND  REFRACTION. 


35 


constitute  an  image  of  the  object.  When  these  conjugate  points 
are  real,  the  image  is  said  to  be  real.  When  the  conjugate 
points  are  virtual,  the  image  is  said  to  be  virtual.  Figure  407 


Fig.  408. 

illustrates  the  formation  of  a  real  image,  and  Fig.  408  that  of 
a  virtual  image. 

647.  Magnification  of  images.  —  The  similarity  of  the  triangles 
aRa1  and  bRV ',  in  Figs.  407  and  408,  shows  that  the  dimensions 
of  the  object  and  of  its 

image  are  proportional  to 
their  respective  distances 
from  the  center  of  cur- 
vature R.  The  ratio  of 
size  of  image  divided  by 
size  of  object  is  called 
the  magnification.  Mag- 
nification is  considered 
negative  when  the  image 
is  inverted. 

648.  Convex   mirrors. 

—  The  action  of  a  con- 

OB 

vex  mirror  is  not  greatly 
different  from  that  of  a 
concave  mirror,  as  will 
be  seen  from  Figs.  409 
and  410.  F*'  410' 


Fig.  409. 


I 

I   IMAGE 


36  ELEMENTS  OF   PHYSICS. 

Figure  409  shows  an  object  and  its  virtual  image  behind  a 
concave  mirror.  Figure  410  similarly  presents  the  case  of  an 
object  and  its  virtual  image  behind  a  convex  mirror.  The  geo- 
metrical construction  in  the  two  cases  is  identical,  excepting 
that  image  and  object  exchange  places. 

649.    Refraction  of  a  plane  wave  at  a  plane  surface.  —  Let 
WW  (Fig.  411)  be  a  plane  wave  approaching  the  surface  AB 


w 


Fig.  411. 

which  separates  two  media,  say  air  and  glass.  Let  the  velocity 
in  glass  be  vt  and  in  air  /-t  times  as  great,  or  pv.  Let  W  W  be 
the  position  which  this  incident  wave  would  reach  at  a  given 
instant  were  it  not  for  AB.  The  envelope  W"  W"  is  the 
refracted  wave,  as  explained  in  Art.  635.  From  the  diagram 

we  have  AB  sin  i  =  d,  (i) 

-*;  (») 


whence 


sin? 
sin  r 


(324) 


REFLECTION  AND  REFRACTION. 


37 


The  angle  between  the  incident  ray  and  the  normal  to  AB 
is  equal  to  the  angle  i  in  Fig.  411;  it  is  called  the  angle  of  inci- 
dence. The  angle  between  the  refracted  ray  and  the  normal  to 
AB  is  equal  to  the  angle  r  in  Fig.  411;  it  is  called  the  angle  of 


refraction.     The  constancy  of  the  ratio 


sm  i 
sinr 


as  expressed  by 


W 


GLASS 


W 


equation  (324),  is  known  as  SnelPs  Law.     This  ratio,  which  is 
equal  to  the  ratio  of  the  velocities  of  the  wave  in  the  respective 
media,  is  called  the  mutual  refractive  index  of  the  two  media. 
The  absolute  refractive  index  of  a  substance 
is  defined  as  the  ratio  of   the  velocity  of 
light    in   vacuo   divided   by  the  velocity  of 
light  in  the  substance.    The  refractive  index 
of  a  substance  varies  with  the  wave  length 
of  the  incident  wave  train,  as  described  in 
the  chapter  on  Dispersion.      Except  when 
stated  to  the  contrary,  the  refractive  index 
of  a  substance  will  be  understood  to  mean 
the    mutual   refractive   index   of    that   sub- 
stance and  air. 

The  angles  between  the  normal  to  the  refracting  surface  and 
the  incident  and  refracted  rays,  satisfy  equation  (324)  (Snail's 
law),  whatever  be  the  shape  of  the  incident  wave  fronts  and 
whatever  be  the  shape  of  the  refracting  surface.  For  let  AB 
(Fig.  412)  be  a  small  portion,  sensibly  plane,  of 
any  refracting  surface,  and  WW  a  small  por- 
tion, sensibly  plane,  of  any  incident  wave.  Then 
WW  will  be  refracted  as  a  plane  wave  at  a 
plane  surface. 


AIR 


Fig.  413. 


650.    Case  of  a  parallel  plate.  —  The  action 
of  a  plate  with  parallel  faces  upon  a  transmitted 
ray  is  shown  in  Fig.  413.     It  will  be  seen  that 
the  incident  ray  and  the  transmitted  ray  are  parallel.      The 
lateral  displacement  (/)  of  the  latter  depends  upon  the  thick- 


ELEMENTS   OF   PHYSICS. 


ness  (/)  of  the  plate,  and  upon  its  refractive  index.     We  have, 

from  Fig.  414, 

t 


t 


1  = 


sin  (i  -  r), 


Fig.  414. 


COS  r 

or  /  =  t  (sin  i  —  cos  i  tan  r). 

Remark.  —  It  is  obvious,  from 
what  has  been  said  about  reflection 
and  refraction,  that  a  ray  turned 
back  upon  itself  will  retrace  its 
path  however  it  may  have  been 
reflected  and  refracted. 


AIR 


651.  The  prism  is  a  portion  of  a  transparent  medium,  e.g. 
glass,  bounded  by  two  polished  surfaces  which  are  inclined  to 

each  other.  A  ray  of  light 
passing  through  a  prism  is  re- 
fracted as  shown  in  Fig.  415. 
The  whole  angle  of  deflection 
a  depends  upon  the  refractive 
index  of  the  prism  upon  the 
angle  <£,  and  upon  the  direc- 
tion of  the  ray  in  the  prism. 

When  the  ray  in  the  prism  is  equally  inclined  to  the  two  faces 

of  the  prism,  the  deflection  a  is  a  minimum.* 

652.  Refraction  of  spherical  waves  at  a  plane  surface.  —  The  line  W  W 
(Fig.  416)  represents  a  wave  front  in  glass  which  has  come  from  a  point  O  in 
air.     The  rays  from  a  small  portion  aa  of  the  wave  W  W  form  an  astigmatic 
pencil  of  which  one  focal  line  is  a  portion  DD  of  the  vertical  line  OV.     The 
other  focal  line  is  at  C,  perpendicular  to  the  paper;    it  lies  on  the  caustic 
(surface).     This  caustic  is  the  evolute  of  an  hyperbola.! 

The  line  W'W  (Fig.  417)  represents  a  wave  front  in  air  which  has  come 
from  a  point  O  in  glass.  The  rays  from  a  small  portion  aa  of  the  wave 
W  W  form  an  astigmatic  pencil  of  which  one  focal  line  is  a  portion  DD  of 
the  vertical  line  OV.  The  other  focal  line  is  at  C,  perpendicular  to  the  paper ; 
it  lies  on  the  caustic  (surface).  This  caustic  is  the  evolute  of  an  ellipse. t 


*  See  Preston's  Light,  p.  100. 


t  Ibid.,  p.  89. 


REFLECTION  AND  REFRACTION. 


39 


GLASS 


GLASS 


Fig.  416. 


AIR 


GLASS 


Fig.  417. 


40  ELEMENTS   OF   PHYSICS. 

653,  Proposition.  —  The  luminous  point  O  (Fig.  417)  is  /x  times  as  far 
from  the  surface  of  the  glass  as  is  the  center  of  curvature  F  of  the  portion 
of  the  wave  near  V. 

Proof.  —  The  center  of  curvature  of  an  advancing  wave  is  stationary  so 
long  as  the  wave  remains  in  the  same  homogeneous  isotropic  medium,  so 

that  it  is  sufficient  to  prove  that  the 
center  of  curvature  of  the  wave  W'W 
(Fig.  417)  is  at  F  immediately  upon  the 
entrance  of  the  wave  into  the  air.  Let 
the  dotted  circle  ivw  (Fig.  418)  be  the 
position  which  would  be  reached  by  a 
wave  from  O  at  a  given  instant  were  it 
not  for  the  discontinuity  from  glass  to 
air;  then 

-ft. 

The  versed  sine  of  the  actual  wave 
in  air  is  /x^,  so  that 

d2 


Therefore  b  =  pa,  which  proves  the  proposition.  In  this  proposition,  h  is 
understood  to  be  small. 

A  similar  proof  shows  that  XF  (Fig.  416)  is  equal  to  /x  times  XO,  where  X 
is  at  the  surface  AB. 


O  Fig.  419. 

Example.  —  The  refractive  index  of  water  is  about  1.33,  so  that  OX 
(Fig.  419)  is  1.33  times  as  great  as  FX.  An  object  O  (Fig.  419)  appears 
to  be  at  F  when  seen  from  V.  When  seen  from  £,  the  point  O  appears  to  be 


REFLECTION  AND  REFRACTION. 


at  D  if  the  head  is  held  erect  so  that  the  line  joining  the  eyes  is  horizontal, 
or  at  C  if  the  head  is  held  so  that  the  line  joining  the  eyes  is  vertical.  This 
is  shown  very  strikingly  by  looking  at  a  bit  of  white  chalk  in  a  dark  vessel  of 
water.  The  apparent  bending  of  a  straight  stick  partially  submerged  in  clear 
water  is  due  to  the  above-described  apparent  displacement  of  the  submerged 
portions. 

654.  Total  reflection.  —  A  ray  of  light  in  passing  from  a 
denser  medium,  such  as  glass  or  water,  into  air  is  bent  away 
from  the  normal.  In  this  case,  since  the  angle  of  refraction 
is  always  greater  than  the  angle  of  incidence,  we  may  write 

sm  l  =  -,  where  JJL  is  the  refractive  index  measured  in  the  usual 

sin  r     ft 

way,  i.e.  by  comparing  glass  with  air  as  a  standard. 

When  the  incident  ray  becomes  so  oblique  that  sin  i  =  -,  we 
have  sin  r  —  i  and  r  —  90°  :  that  is,  the  ray  in  air  is  parallel  to 
the  surface.  When  sin  i  is  greater  than  — ,  the  law  of  refraction 

would  require  sin  r  to  be  greater  than  unity,  which  is  impossible  ; 
and  Huygens'  construction  for 
the  refracted  wave  front  (Art. 
649)  entirely  fails.  In  fact,  the 
light  is  totally  reflected.  This 
is  shown  in  Fig.  417,  for  which 
fji  =  1.50.  The  rays  from  O, 
which  strike  that  portion  of  the 
surface  which  is  marked  by  the 
fine  line,  are  partially  transmit- 
ted. The  rays  striking  beyond 
this  portion  are  totally  reflected. 
The  well-known  lecture  ex- 
periment of  the  illuminated 
water  jet  is  based  upon  the 
phenomenon  of  total  reflection. 

Fig.  420. 

In   this   experiment  water   es- 
capes from  a  small  circular  orifice,  o,  near  the  base  of  a  tall 
cylindrical  tank,  shown  in  Fig.  420. 


42  ELEMENTS   OF   PHYSICS. 

In  the  opposite  wall  of  the  tank,  and  at  the  same  level,  there 
is  a  lens  forming  a  window,  w,  through  which  a  beam  of  light 
enters  the  tank  and  is  concentrated  in  the  orifice  o.  The  beam 
of  light  reaches  the  wall  of  the  jet  at  grazing  incidence  and 
is  totally  reflected.  From  this  point  on  it  is  totally  reflected 
whenever  it  meets  the  outer  wall  of  the  jet,  and  is  thus  con- 
veyed, almost  without  loss,  along  the  parabolic  path  of  the  latter 
for  its  whole  length.  The  fine  particles  of  dust  in  the  jet 
diffuse  the  beam  of  light,  causing  the  whole  jet  to  be  softly 
luminous.  In  a  similar  manner  the  field  of  a  microscope  may 
be  illuminated  from  a  lamp  placed  at  a  distance  in  any  con- 
venient position.  The  beam  of  light  is  conveyed  the  entire 
length  of  a  glass  rod,  one  end  of  which  is  beneath  the  slide 
while  the  other,  which  is  plane  and  polished,  is  near  the  lamp 
flame.  The  rod  may  be  bent  into  almost  any  form. 

655.   Refraction  at  a  spherical  surface. 

(a)  General  case.  —  A  wave,  plane  or  spherical,  when  re- 
fracted at  a  spherical  surface,  is,  in  general,  no  longer  spherical. 
There  is,  however,  a  particular  case  discussed  in  Art.  657  in 

which  the  refracted  wave  is  spherical. 

(b)   Case  of  the  refraction  of  a  nar- 

AIR  OR         '/  .,     -  . 

GLASS     fsfi  row  pencil  of  rays  at  a  portion  of  a 

spherical  surface  which  is  small  com- 
pared with  its  radius  of  curvature.  — 
In  this  case  the  refracted  pencil  is 
sensibly  homocentric  when  the  inci- 
dent pencil  falls  nearly  perpendicu- 
Fig  42j  larly  upon  the  refracting  surface. 

Such  a  pencil  is  shown  in  OV,  Fig. 

421.     If  the  incidence  is  oblique,  as  in  the  case  of  the  beam 
OX,  the  refracted  pencil  is  distinctly  astigmatic. 

Consider  a  small  portion  AB  (Fig.  422)  of  a  refracting  surface  of  which 
the  radius  of  curvature  is  R,  and  center  of  curvature  at  R.  It  is  easy  to  show, 
by  the  method  of  Art.  653,  that  plane  waves  from  the  right  are  concentrated 


REFLECTION  AND  REFRACTION.  43 

at  /'"at  a  distance R  from  AB,  and  that  plane  waves  from  the  left  are 

/x. 

concentrated  at  F  at  a  distance      _  ;  /?  from  AB.     Therefore  any  ray  par- 
allel to  the  axis  passes  through  F  or  F' ,  after  reflection,  according  as  the  ray 

GLASS 


Fig.  422. 

is  from  the  right  or  from  the  left.  Since,  also,  a  ray  passing  through  the 
center  of  curvature  crosses  AB  perpendicularly,  and  is  unchanged  in  direction, 
the  statements  of  Arts.  644  to  647  hold  for  the  present  case. 

656.  Spherical    aberration ;     aplanatic    surface.  —  When    a 
spherical  wave  is  no  longer  spherical  after  reflection  or  refrac- 
tion, it  is  said  to  have  suffered  spherical  aberration.     When  a 
spherical  wave  remains  spherical,  after  refraction,  at  a  surface, 
e.g.  between  glass  and  air,  the  refracting  surface  is  said  to  be 
aplanatic. 

657.  Refraction  at  a  spherical  surface  without  spherical  aber- 
ration.— The  one  case  in  which  the  spherical  surface  is  aplanatic 
is  as  follows :  Consider  two  points  P  and  P'  (Fig.  423).     Choose 

P' C     PfCf 
the  points  C  and  C'  so  that  — —  =  — —  =  fi,  and  describe  a 

JL     Ls  JT  Ls 

circle  (sphere)  on  CC'  as  a  diameter.  Then  by  geometry  each 
point  of  this  sphere  is  //,  times  as  far  from  P'  as  it  is  from  P. 
Imagine  the  sphere  to  be  made  of  glass  of  refractive  index  //,, 
and  let  P  be  a  luminous  point.  A  wave  of  light  from  P  is 
spherical  after  passing  into  the  air,  and  its  center  of  curvature 
is  P1. 

Proof. — Let  the  dotted  circle  WW  of  radius  r  be  the  posi- 
tion which  a  wave  from  P  would  reach  at  a  given  instant  if  the 
whole  region  were  glass.  Describe  a  circle  W1  W1  of  radius  pr 
with  its  center  at  P' .  Consider  the  wavelet  from  the  point  /, 
distant  b  from  P  and  ^b  from  P' .  At  the  given  instant  this 


44 


ELEMENTS   OF   PHYSICS. 


wavelet  has  had  time  to  travel  a  distance  (r  —  b]  in  glass  or 
p(r—b)  in  air,  so  that  its  radius  is  p(r—  b).     Therefore,  since 


AIR 


W 


Fig.  423. 


p(r—b)+nb  =  pr,  this  wavelet  touches  W  W  ;  W  W1  is  thus 
the  envelope  of  all  such  wavelets,  and  therefore  it  is  the  re- 
fracted wave.  Q.E.D. 


CHAPTER   IV. 

LENSES. 

658.  The  lens  is  a  portion  of  a  transparent  medium,  generally 
glass,  bounded  by  polished  spherical  surfaces.     Figures  424  and 
425  show  two  typical  cases.    The 

line   RR'   passing    through    the 

centers  of  curvature  of  the  spher- 

ical  surfaces  is  called  the  axis  of 

the  lens.    A  lens  which  is  thicker 

at  the  center  than  at  the  edge  is  Fig'  424' 

called  a  converging  lens.    A  lens  which  is  thinner  at  the  center 

than  at  the  edge  is  called  a  diverging  lens. 

A  plane  or  spherical  wave  is,  in  general,  no  longer  spherical 
after  passing  through  a  lens.  If  the  lens  is  very  thin  compared 
with  its  diameter,  or  if  the 
aperture  *  of  the  lens  is 

D 

small,  and  if  the  refracted  — -x 
wave  has  come  from  a 
point  in  or  near  the  axis 
of  the  lens,  then  the  wave 
after  passing  is  sensibly  spherical.  The  discussion  in  the  fol- 
lowing articles  refers  to  thin  lenses.  Thick  lenses  of  small 
aperture  are  treated  as  lens  systems. 

659.  Principal  focus.     Principal  focal  length.  —  The  points 
(one  on  either  side)  at  which  rays  parallel  to  the  axis  are  con- 
centrated by  a  lens  are  called  the  principal  foci.     In  case  of  a 

*  That  is,  if  only  the  central  portion  of  a  large  lens  is  used. 

45 


46 


ELEMENTS   OF   PHYSICS. 


diverging  lens,  parallel  rays  seem  to  have  come  from  a  principal 
focus  after  passing  through  the  lens.  The  distance  of  the  prin- 
cipal foci  from  a  lens  is  called  the  principal  focal  length  of  the 
lens.  Figure  426  shows  a  principal  focus  of  a  converging  lens, 
and  Fig.  427  shows  a  principal  focus  of  a  diverging  lens. 


Fig.  426. 

660.    Proposition.  —  The  principal  focal  length  p  of  a  lens  is 

(325) 


in  which  /*  is  the  refractive  index  of  the  glass  referred  to  air, 
and  R  and  R'  are  the  radii  of  curvature  of  the  surfaces  of  the 
lens.* 


Fig.  427. 


Proof. — Let  d  be  the  diameter  of  the  lens,  measured  from  its 
true  edge  where  the  faces  intersect.  Imagine  a  plane  drawn 
through  this  thin  edge,  and  let  h  and  //'  be  the  distances  repre- 
sented in  Fig.  424. 

*  Either  radius,  R  or  R',  is  considered  negative  when  the  corresponding  surface 
is  concave;  and  the  focal  length/  is  considered  negative  when  the  lens  is  diverging. 


LENSES. 

Then  *- 


and 


The  thickness  of  the  lens  at  its  center  is  h  +  h'.  While  the 
central  part  of  a  plane  wave  is  traveling  this  distance  through 
glass,  the  portions  of  the  wave  which  graze  the  edge  of  the  lens 
travel  /JL  times  as  far  in  air,  and  gain  on  the  central  part  of  the 
wave  by  the  amount  p  (h  +  h')  —  (h  -f-  k')  or  (/JL  —  i)(k  +  h'). 
This  quantity  is  the  versed  sine  of  the  transmitted  wave  of 
which  the  chord  is  d,  so  that 

I  f=s(p-w  +  *>j  (326) 

Substituting  the  values  of  h  and  h'  from  (i)  and  (ii)  in  (326), 
d  drops  out  also,  and  we  have  equation  (325).  Q.E.D. 

Remark  i.  —  Any  wave  in  passing  through  a  lens  has  its 
central  portion  retarded  by  the  amount  (/*,  —  i)  (h  +  h'). 

Remark  2.  —  The  above  proof  may  be  adapted  to  a  diverging 
lens  if  we  think  of  the  lens  coming  to  infinitesimal  thickness  at 
the  center,  and  having  a  thickness  h  +  k'  at  the  edge. 

Remark  3.  —  The  value  of  //,,  and  therefore  of  /  also,  varies 
with  the  wave  length  of  the  light  (Art.  676  on  chromatic  aberra- 
tion). In  the  following  discussion  p  and/  are  assumed  to  have 
definite  values. 

661.  Conjugate  foci.  —  Two  points  so  situated  that  light  from 
one,  after  passing  through  a  lens,  is  concentrated  at,  or  appears 
to  have  come  from,  the  other  are  called  conjugate  foci  or  conju- 
gate points. 

Proposition.  —  The  distances  a  and  b  of  a  pair  of  conjugate 
points  from  the  center  of  a  lens,  and  the  principal  focal  length 
p,  satisfy  the  equation 


48 


ELEMENTS   OF   PHYSICS. 


Proof.  —  Consider  a  portion  of  a  wave,  which  has  reached  the 
lens  from  a  distance  a,  of  which  the  chord  is  equal  to  the 
diameter  of  the  lens  d,  and  of  which  the  versed  sine  is  k ;  then 


a  = 


The  action  of  the  lens  is  to  retard  the  central  portion  of  this 
wave  by  the  amount  (/*  —  i)(/z  -h  h')  [see  Art.  660,  Remark  i], 
so  that  the  versed  sine  of  the  transmitted  wave  is 


and  its  radius  of  curvature  b  is 


(ii) 


Substituting  the  values  of  /,  a,  and  b  from  equation  (326)  of 
Art.  660,  and  (i)  and  (ii)  in  equation  (327),  we  find  that  equation 
to  be  identically  satisfied.  Q.E.D. 

662.  Conjugate  points  out  of  axis.  —  A  pair  of  conjugate 
points  which  do  not  lie  on  the  axis  of  a  lens,  lie  on  a  straight 
line  passing  through  the  center  of  the  lens. 

Proof.  —  Consider  a  ray  from  a  point  a  (Fig.  428)  passing 
through  the  center  of  a  lens.  This  ray  passes  through  the  two 
surfaces  of  the  lens  at  points  where  these  surfaces  are  parallel. 


Fig.  428. 


Therefore  the  lens  acts  upon  this  ray  as  a  thin  parallel  plate, 
and  the  emergent  ray  is  sensibly  a  continuation  of  the  incident 


LENSES. 


49 


ray.  (Compare  Art.  650.)  This  emergent  ray  passes  through 
the  point  conjugate  to  the  point  a.  Therefore  the  proposition 
is  proven. 

Geometrical  construction  for 
the  conjugate  of  a  point.  —  Draw 
a  line  (ray)  from  the  point  a 
(Fig.  428  or  429)  through  the 
center  of  the  lens.  Draw  a  line 
(ray)  from  a  parallel  to  the  axis 
of  the  lens,  and  from  the  inter- 
section of  this  ray  with  the  lens, 
which  is  supposed  thin,  draw  a  line  through  the  principal  focus 
F.  Then  the  point  b  is  the  conjugate  of  a. 

The  statements  of  Arts.  645,  646,  and  647  hold  for  the  case  of 

size  of  object 

a  lens.     Ine  magnification  01  an  imasce,  that  is,  — •=—. — - — > 

size  of  image 

is  equal  to  the  ratio  of  the  distances  of  object  and  image  from 
the  center  of  the  lens. 


Fig.  429. 


663.  Centered  system  of  lenses.  —  Consider  a  number  of  trans- 
parent media,  A,  B,  C,  D>  E,  F  (Fig.  456),  for  example,  air  and 
various  kinds  of  glass,  separated  by  spherical  surfaces  of  which 
the  centers  lie  on  a  straight  line,  —  the  axis.     Such  an  arrange- 
ment is  called  a  centered  lens  system.     A  system  consisting  of 
two  lenses  is  called  a  doublet;    a  system   consisting  of  three 
lenses  is  called  a  triplet. 

664.  Propositions.  —  (a)  A  narrow  pencil  of  rays  from  any  luminous  point 
O,  in  or  near  the  axis  *  of  a  lens  system  is  sensibly  homocentric  after  passing 
through  the  system,  and  is  concentrated  at  or  appears  to  have  come  from  a 
point  O',  called  the  conjugate  of  O. 

Proof.  —  According  to  Art.  655,  a  narrow  pencil  near  the  axis  will  be  homo- 
centric  after  refraction  at  the  first  spherical  surface.  The  resulting  homo- 
centric  pencil  will  be  homocentric  after  refraction  at  the  second  spherical 
surface,  and  so  on. 

*  In  the  following  figures  rays  are  drawn  which  make  considerable  angles  with 
the  axis  for  the  sake  of  clearness. 


OF  THK 

IVERSITY 


ELEMENTS   OF   PHYSICS. 


(£)  A  group  of  luminous  points  (an  object)  near  the  axis  in  a  plane 
perpendicular  to  the  axis  has  as  its  image  (with  definite  magnification,  posi- 
tive or  negative)  a  similar  group  of  points  near  the  axis  in  another  plane 
perpendicular  to  the  axis.  Such  planes  are  called  conjugate  planes. 

Proof.  —  By  Art.  655  an  object  has  an  image  formed  in  accordance  with 
this  proposition  by  the  first  refracting  surface.  An  image  of  this  image  is  pro- 
duced by  the  second  refracting  surface,  and  so  on.  'Q.E.D. 

Corollary  i .  —  Any  incident  ray  passing  through  a  point  O  passes  upon 
emergence  through  O',  the  conjugate  of  O.  For,  by  proposition  («),  all  rays 
emanating  from  or  passing  through  O  pass  through  O'  upon  emergence.  An 
incident  ray  and  its  position  upon  emergence  are  called  conjugate  rays. 

Corollary  2 .  —  Two  incident  rays  intersecting  at  O,  intersect  upon  emer- 
gence at  O'. 

Corollary  3.  —  Consider  an  incident  ray  passing  through  the  points  O  and^. 
This  ray  upon  emergence  passes  through  O'  and^'. 

Remark  i .  —  If  an  emergent  ray  be  reversed,  it  will  retrace  its  path,  as 
explained  in  Art.  650.  If,  therefore,  the  ray  r'  is  the  conjugate  of  r,  then  r 
is  the  conjugate  of  r' ;  and  if  the  point  O'  is  the  conjugate  of  <9,  then  O  is 
the  conjugate  of  O'. 

Remark  2. —  Any  specification  which  fixes  the  positions  upon  emergence 
of  given  incident  rays  is  a  complete  specification  of  the  lens  system. 

665.  Specification  of  lens  systems.  —  The  action  of  a  lens  system  is  com- 
pletely specified  when  the  positions  of  two  pairs  of  conjugate  planes  are  given, 
together  with  the  magnification  associated  with  each  pair. 

Proof.  —  Let  aa'  and  bb*  (Fig.  430)  be  two  given  pairs  of  conjugate  planes. 

Let  the  magnification  —  be  m,  and  the  magnification  —  be  m'.    Given  an  inci- 
a  b' 

dent  ray  r'  (from  the  left),  cutting  the  planes  a'  and  V  at  p'  and  q',  as  shown. 


md 


md 


Fig.  430. 


The  transmitted  ray  r  must  pass  through  the  points  p  and  q,  which  are  con- 
jugate to  p'  and  q'  respectively.  The  transmitted  ray  is  thus  fixed,  and  the 
action  of  the  lens  system  upon  the  ray  r1  is  completely  determined  by  the 
given  data.  Q.E.D. 


LENSES.  5 1 

666.  Principal  foci  of  a  lens  system.  —  Consider  an  incident  ray  r'  (Fig. 
431),  from  the  left,  parallel  to  the  axis.  The  conjugate  ray  r  is  as  shown. 
If  the  distance  d  changes,  the  distances  md  and  m'd  change  in  the  same  ratio, 


Fig.  431. 

as  shown  by  the  dotted  rays  R  and  R',  and  the  point  F  remains  fixed.  There- 
fore all  rays  from  the  left,  parallel  to  the  axis,  pass  through  F,  or  seem  to 
have  come  from  F  after  emergence.  This  point  Fis  called  the  (right)  princi- 
pal focus  of  the  system.  Figure  432  shows  the  construction  for  the  left  prin- 
cipal focus  F'.  ^V 


a' 

b' 

b 

a 

r 

r> 

/ 

R 

s^tf 

AXIS 

>- 

'^' 

/ 

w  1^ 

W/=  +1^ 

Fig.  432. 


667.  Principal  planes  of  a  lens  system. — That  pair  of  conjugate  planes 
for  which  the  magnification  is  plus  one  (+  i)  are  called  the  principal  planes 
of  the  system.  The  following  discussion  shows  that  there  is  always  such  a 
pair  of  planes,  and  shows  their  location  relative  to  the  given  conjugate  planes 
aa'  and  bb' . 

Let  r  (Fig.  433)  be  an  incident  ray  from  the  right,  and  r'  its  conjugate. 
Let  R',  colinear  with  r,  be  an  incident  ray  from  the  left,  and  R  its  conjugate. 
The  rays  r  and  R  intersect  at  O,  and  the  rays  r'  and  R',  conjugate  to  r  and 
R.  intersect  at  O'.  Therefore  O  and  O1  are  conjugate  points,  and  since  they 
are  at  the  same  distance  from  the  axis,  it  follows  that  the  magnification  for 
the  conjugate  planes  PP  is  unity,  and  that  P  and  P  are  the  required  principal 
planes  of  the  system.  The  distance  Fto  Pis  called  the  right  focal  length  of 
the  system,  and  the  distance  F1  to  P1  the  left  focal  length  of  the  system.  The 
system  represented  in  Figs.  430,  431,  432,  433,  435,  and  436  is  a  converging 


52  ELEMENTS   OF  PHYSICS. 

system.  The  two  focal  lengths  of  a  lens  system  have  a  ratio  equal  to  the 
ratio  of  the  refractive  indices  of  the  media,  in  which  the  principal  foci  are 
situated.  The  simplest  case  is  that  described  in  Art.  655.  In  most  lens  sys- 
tems used  in  practice  the  two  focal  lengths  are  equal. 


Fig.  433. 

668.  Example. — Figure  434  shows  to  scale  the  actual  positions  of  the 
principal  planes  /*and  P,  and  of  the  principal  foci  Fand  F  of  a  symmetrical 
bi-convex  lens  —  not  of  infinitesimal  thickness  —  the  glass  of  which  has  a  re- 
fractive index  of  1.50.  The  figure  also  shows  the  geometrical  construction 
for  determining  the  position  of  the  image  of  an  object.  Draw  the  ray  r 


Fig.  434. 

parallel  to  the  axis  from  O  to'  the  principal  plane  /",  thence  after  emergence 
through  the  focus  F1 '.  Draw  the  ray  R  from  F  through  O  to  the  principal 
plane  P,  thence  after  emergence  parallel  to  the  axis.  The  conjugate  of  O  is 
at  the  intersection  of  r'  and  R'.  It  is  marked  O'. 

669.  The  inverse  principal  planes  of  a  lens  system  are  conjugate  planes 
for  which  the  magnification  is  minus  one  (—  i).  The  following  discussion 
shows  that  there  is  always  such  a  pair  of  conjugate  planes,  and  shows  their 


LENSES. 


53 


location.  Let  the  principal  planes  P  and  P'  and  the  foci  F  and  F'  (Fig.  435) 
be  given.  This  constitutes  a  complete  specification  of  the  system.  Let  R  and 
R1  be  conjugate  rays.  Consider  the  point  q',  which  is  on  the  ray  R'  and  at  a 
distance  d  below  the  axis.  The  conjugate  of  q'  must  lie  on  the  ray  R  at  a  dis- 
tance d  above  the  axis.  Therefore,  the  plane  Qf,  passing  through  the  point  g',  is 
one  of  the  inverse  principal  planes.  To  determine  the  other  inverse  principal 


Q' 


P' 


--,~-*r- r-t *K: 

F'R^ 


Fig.  435. 

plane,  consider  the  conjugate  rays  r'  and  r.  The  point  ^  is  the  conjugate  of 
g',  and  the  plane  Q  is  the  other  inverse  principal  plane.  From  Fig.  435,  it  is 
evident  that  the  distance  PQ  is  2/,  and  that  the  distance  P'Q'  is  2/' ;  where 
/  and  /'  are  the  right  and  left  focal  lengths  of  the  lens  system  respectively. 

670.  The  nodal  points  of  a  lens  system  are  the  two  conjugate  points  in 
the  axis  such  that  any  ray  passing  through  them  is,  upon  emergence,  parallel 
to  its  direction  upon  incidence.  Consider  an  object  in  the  plane  Q  (Fig.  435) 
and  its  inverted  image  in  the  plane  Q' .  Imagine  a  ray  r  (not  shown  in  the 
figure)  passing  from  the  point  q  to  the  nodal  point  N  (not  shown)  and  its 
conjugate  *  ray  r'  passing  from  the  nodal  point  N'  to  q' '.  The  image,  being  of 
same  size  as  the  object  and  inverted,  and  r  being,  by  definition  of  nodal  points, 
parallel  to  r',  it  must  be  that  N  and  N'  are  at  equal  distances  and  in  opposite 
directions  from  Q  and  Q'  respectively.  Similarly,  N  and  N'  are  at  equal 
distances  and  in  the  same  direction  from  P  and  P'  respectively,  as  shown  in 
Fig.  436.  Let  x  be  the  distance  of  N  and  N'  to  the  left  of  P  and  /",  and  y  the 
distance  of  N  and  N'  from  Q  and  Q.  Then,  since  Q'P'  =  2/'  and  QP  =  2/, 

we  have  x-\-  y  =  if 

and  y  —  x  =  2f 

whence  x—f  —  f\ 


*  It  being  assumed  that  N  and  N'  are  conjugate  points. 


54 


ELEMENTS   OF   PHYSICS. 


It  remains  to  be  shown  that  N  and  N',  the  positions  of  which  are  determined 
by  (ii),  are  conjugate  points.  Draw  any  parallel  lines  r  and  r'  (Fig.  436) 
through  jVand  N' .  Then,  since  NQ  =  N'Q'  and  NP  =  N' F ,  we  have  e  =  e' 
and  d  =  d ',  so  that  the  points  p'  and  q'  are  the  conjugates  of  the  points  p  and  q  ; 
and  r  and  r'  are  conjugate  rays.  The  same  would  be  true  of  another  pair  of 


Fig.  436. 

parallel  lines  R  and  R'  (not  shown)  passing  through  N  and  N'.  R  and  r 
intersect  at  N",  and  their  conjugate  rays  R'  and  r'  intersect  at  N',  so  that  N 
and  N'  are  conjugate  points. 

Corollary.  —  Any  object  and  its  image  subtend  equal  angles,  as  seen  from 
the  respective  nodal  points  of  a  lens  system. 

Remark.  —  When  the  right  and  left  focal  lengths  of  a  system  are  equal,  the 
nodal  points  lie  in  the  principal  planes  P  and  P ' . 


CHAPTER   V. 
THE   CORRECTION   OF   LENSES;    LENS   SYSTEMS. 

671.  General  statement.  —  The  elementary  theory  given   in 
Chapter  IV.  applies  to  thin  lenses,  and  to  mirrors  and  centered 
lens  systems  of  small  aperture.     It  assumes  all  imaged  points 
to  be  nearly  in  the  axis  and  the  refractive  index  of  a  sample  of 
glass  to  be  the  same  for  all  kinds  of  light.     These  conditions 
are  not  realized  in  practice,  and  lenses  have  therefore  certain 
imperfections.    These  imperfections  may  be  very  largely  avoided 
by  using  properly  designed  lens  systems  instead  of  simple  lenses. 

The  approach  to  perfection  attained  in  the  manufacture  of 
lenses  depends  upon  the  fact  that  the  resolving  power  of  the  eye 
is  small.  Lenses  are,  in  general,  used  to  aid  vision,  directly  or 
indirectly.  In  order  to  secure  a  satisfactory  result,  the  region 
within  which  the  light  from  a  point  of  an  object  is  concen- 
trated by  a  lens  must  be  so  small  as  to  be,  under  the  conditions 
in  which  it  is  viewed,  sensibly  a  point. 

Remark.  —  It  is  a  familiar  fact  that  we  can  see  distinct  images,  more  or  less 
distorted,  of  surrounding  objects  in  almost  any  polished  surface,  however 
uneven,  and  that  objects  are  not  greatly  confused  when  seen  through  window 
glass  and  through  smooth  glassware.  This  is  due  to  the  fact  that  in  most 
cases  the  portion  of  such  a  surface  which  sends  light  into  the  eye  from  a 
given  luminous  point  is  very  small,  much  smaller  even  than  the  pupil  of  the 
eye.  The  focal  lines  of  a  narrow  astigmatic  pencil  are  very  short  when  they 
are  near  together,  and  if  these  short  focal  lines  are  at  a  considerable  distance 
from  the  eye,  the  astigmatic  pencil  is  sensibly  homocentric. 

672.  Definitions.     Numerical  aperture.  —  The   effective   free 
diameter  of  a  lens  divided  by  its  principal  focal  length  is  called 
its  numerical  aperture. 

55 


56  ELEMENTS   OF   PHYSICS. 

Telescopic  objectives  range  up  to  y1^,  photographic  objectives 
up  to  J  or  J,  and  microscopic  objectives  up  to  1.4  numerical 
aperture. 

Field  angle.  —  The  angle  at  the  center  of  a  lens,  between 
lines  drawn  to  the  extreme  edges  of  the  largest  distinct  image 
which  the  lens  will  produce,  is  called  the  field  angle  of  the  lens. 
A  lens  so  designed  as  to  have  a  wide  field  angle  is  called  a 
wide  angle  lens.  Photographic  objectives  are  made  which  give 
excellent  definition  with  a  field  angle  as  great  as  110°.  The 
field  angle  of  telescopic  and  microscopic  objectives  is  ordinarily 
very  small,  seldom  exceeding  5°. 

673.  Spherical    aberration.  —  When    a    plane   (or   spherical) 
wave  passes   through  a  simple   lens  of   wide   aperture,   those 
portions  of  the  wave  which  pass  through  the  outer  zones  of 
the  lens  are  focused  nearer  the  lens  than  those  portions  are 

which  pass  through  the  central  zone. 
(See  Fig.  437.)  In  other  words,  the 
outer  zones  of  a  lens  are  of  shorter 
focal  length  than  the  central  zone. 
This  fault  of  lens  is  called  spherical 
aberration.  A  lens  free  from  spheri- 

Fig.  437. 

cal  aberration  is  said  to  be  aplanatic. 

The  spherical  aberration  of  a  simple  converging  lens,  as  also  its 
focal  length,  is  considered  to  be  positive.  Of  a  simple  diverg- 
ing lens  both  are  considered  to  be  negative.  The  greater  the 
refractive  index  of  the  glass  used,  the  less  the  thickness  of  a 
lens  of  given  focal  length,  and  the  less  its  spherical  aberration. 

674.  Correction  of  spherical  aberration.  —  A  converging  lens 
and  a  diverging  lens  of  equal  and  opposite  focal  lengths,  and 
of  equal  and  opposite  spherical  aberration,  entirely  annul  each 
other's  action  if  placed  near  together.     The  spherical  aberration 
of  a  lens  of  a  given  focal  length  depends,  however,  upon  the 
relative  curvature  of  the  two  faces  of  the  lens  and  upon  the 
refractive  index  of  the  glass  ;  so  that  it  is  possible  to  construct 


LENS    SYSTEMS. 


57 


a  converging  lens  and  a  diverging  lens  giving  equal  and  oppo- 
site spherical  aberration,  but  not  having  equal  and  opposite  focal 
lengths.  When  two  such  lenses  are  used  together,  they  form 
an  aplanatic  doublet  of  any  required  focal  length.  A  lens 
system  can  be  aplanatic  only  for  a  given  pair  of  conjugate 
points  (planes)  called  the  aplanatic  points  of  the  system. 

675.  Abbe's  condition  for  aplanatism.  —  If  a  lens  is  to  form  a  distinct 
image  of  a  small  group  of  points  in  and  near  its  axis  at  one  of  its  aplanatic 
points,  the  lens  must  be  sensibly  aplanatic  for  the  points  near  the  axis  as  well 
as  for  the  point  in  the  axis.  The  condition  that  must  be  satisfied  in  order 
that  this  may  be  true  is  called,  from  its  discoverer,  Abbe's  Sine  Law. 

Preliminary  statement.  —  Let  a  and  b  (Fig.  438)  be  two  conjugate  points 
with  respect  to  a  lens  system.  A  spherical  wave  passing  out  from  a  becomes 
a  spherical  wave  coming  in  upon  b.  Let  w  and  w'  be  the  portions  of  this 
spherical  wave  which  have  traveled  along 
any  two  rays  r  and  r'.  Since  iv  and  w' 
started  out  from  a  at  the  same  instant, 
and  are  portions  of  a  spherical  surface 
with  its  center  at  £,  they  will  reach  b  at 
the  same  instant.  It  follows,  therefore, 
that  the  time  required  for  a  light  wave  to  travel  along  any  two  rays  from 
a  point  to  its  conjugate  is  the  same.  This  time  is  also  the  least  time  in  which 
light  can  pass  from  one  point  to  the  other.  This  principle  is  sometimes  called 
the  principle  of  least  time. 

Consider  a  very  small  object  aa'  and  its  image  bb'  (Fig.  439 a}.  Let  d  be 
the  distance  aa1  and  md  the  distance  bb',  m  being  the  magnification.  Con- 


Fig.  438. 


Fig.  439  (a). 


sider  the  rays  R  (the  axis)  and  R'  from  a  to  b  and  r  and  r'  from  a'  to  b' . 
Let  <J>  be  the  angle  between  the  incident  portions  of  R'  and  r'  and  the  axis 
(these  angles  are  sensibly  equal)  ;  and  let  3>  be  the  angle  between  the  emer- 
gent portions  of  R'  and  R  and  of  r'  and  r  (these  angles  are  sensibly  equal) . 

The  rays  R  and  r  pass  through  the  same  portion  of  the  lens,  and  are  of 
the  same  length,  so  that  the  rays  r'  and  R'  must  be  of  the  same  (optical) 
length  also.  The  rays  r'  and  R'  pass  through  the  same  thickness  of  glass, 


58  ELEMENTS   OF   PHYSICS. 

the  incident  portion  of  R'  is  dsin(j>  longer  than  the  incident  portion  of  r', 
and  the  emergent  portion  of  r'  is  mdsin&  longer  than  the  emergent  portion 
of  R',  as  shown  in  Fig.  439  £.  Therefore,  dsin  <j>  = 

sinrf) 

r;  - 


(328) 


If  the  refractive  indices  of  the  media  in 
which  the  object  and  image  are  located  are  /A 
and  p!  respectively,  then  the  optical  values  of 


Fig.  439  (b). 


dsin  <(>  and  mdsin  $  are 
and  equation  (328)  becomes 


and 


sin  $ 


mdsin  <I> 


(329) 


The  reader  will  find,  by  applying  equation  (329),  that  the  sphere  described 
in  Art.  657  satisfies  Abbe's  Sine  Law,  and  is  therefore  aplanatic  for  all  points 
near  the  axis  in  a  plane  passing  through  the  point  P  (Fig.  423)  perpendicular 
to  the  axis.  If  this  were  not  the  case,  a  microscope  provided  with  the 
objective,  shown  in  Fig.  450,  would  give  sharp  definition  only  in  the  very 
central  point  of  the  field  of  view. 

676.  Chromatic  aberration.  —  The  value  of  the  refractive 
index  of  a  given  sample  of  glass  varies  with  the  wave  length  of 
the  incident  wave  train.  Therefore  (see  Art.  660)  the  focal 
length  of  a  lens  is  different  for  different  wave  lengths  (colors). 
Violet  light  is  focused  nearer  the  lens  than  red  light  and  the 
intermediate  colors  between,  as  shown  at  r  and  v  (Fig.  440). 
This  variation  of  focal  length  of  a  lens  with  wave  length  is 
called  chromatic  aberration.  It  is  possible  to  construct  a  con- 
verging lens  and  a  diverging 
lens,  giving  equal  and  oppo- 
site chromatic  aberration  (for, 
r  say,  red  and  violet  light),  but 
having  different  focal  lengths, 
so  that  when  used  together  they  will  form  a  doublet  of  any 
desired  focal  length.  This  doublet  will  have  equal  focal  lengths 
for  the  two  colors.  Such  a  lens  system  is  called  an  achromatic 
lens.  A  sketch  of  the  elementary  theory  of  the  achromatic 
lens  is  given  in  the  chapter  on  Dispersion. 


Fig.  440. 


LENS   SYSTEMS. 


VERSITZ 


59 


A  lens  system  consisting  of  two  thin  lenses  of  similar  glass 
at  a  distance  apart  equal  to  half  the  sum  of  their  individual 
focal  lengths  is  achromatic.*  The  doublets  of  Ramsden  and 
Huygens,  illustrated  in  Figs.  448  and  449,  are  examples. 

677.  Astigmatism.  —  A  pencil  of  parallel  rays  (in  general  any 
homocentric  pencil)  becomes  an  astigmatic  pencil  when  it  passes 
obliquely  through  a  simple  lens.  Figure  441  shows  the  actual 
positions  of  the  focal  lines,  C  and  DD,  of  the  pencil  RR  of 
parallel  rays  after  passing  through  a  lens  of  which  the  principal 
focal  length  is  OP,  and  of  which  the  glass  has  a  refractive 
index  equal  to  f.  The 
astigmatism  of  a  converg- 
ing lens  being  considered 
positive,  that  of  a  diverg- 
ing lens  is  to  be  consid- 
ered negative.  The  as- 
tigmatism of  a  thin  lens 

Fig.  441. 

of  given  focal  length  in- 

creases with  the  refractive  index  of  the  glass  of  which  the 
lens  is  made.  It  is,  therefore,  possible  to  construct  a  converg- 
ing lens  and  a  diverging  lens,  giving  (to  first  order  of  small 
quantities)  equal  and  opposite  astigmatism,  but  having  different 

*  Proof.  —  It  can  be  shown  that  the  focal  length,/  of  a  doublet  is  such  that 

1     I     I      D 


in  which  /i  and  /2  are  the  respective  focal  lengths  of  the  constituent  simple  lenses, 
and  D  is  their  distance  apart.  Let  A/i  and  A/a  be  the  difference  in  the  values  of  /i 
and/,  respectively,  for  two  colors.  Then  since  the  lenses  are  of  similar  glass,  we  have 


- 

/i  '/a 

The  change  A/  in  the  focal  length  of  the  doublet  due  to  the  changes  A/i  and  A/2 
is  easily  found  from  (i)  .  Placing  this  expression  for  A/"  equal  to  zero,  A/i  and  A/2 
may  be  eliminated  with  the  help  of  (ii),  giving  at  once 

/>=.A±A  Q.E.D.     (330) 


6o 


ELEMENTS   OF   PHYSICS. 


focal  lengths,  so  that  when  used  together  they  may  give  a 
doublet  of  any  desired  focal  length,  sensibly  free  from  astig- 
matism. Such  a  lens  is  called  an  anastigmatic  lens. 

678.  Distortion. — The  image  of  an  object  formed  by  a  lens 
is,  in  general,  distorted. 

In  case  the  magnification  increases  towards  the  edge  of  the 
field,  the  image  of  a  square  network  will  appear,  as  Fig.  442  a. 


Fig.  442. 

In  case  the  magnification  decreases  towards  the  edge  of  the  field, 
the  image  of  a  square  network  will  appear,  as  Fig.  442  c.  In 
case  the  magnification  is  constant,  the  image  will  not  be 
distorted. 

A  lens  which  does  not  give  a  distorted  image  of  an  object  is 
called  an  orthoscopic  or  rectilinear  lens.  The  simplest  ortho- 
scopic  combination  is  the  symmetrical  doublet  first  introduced 

by  Steinheil.  This  consists  of  a 
combination  of  two  similar  lenses 
LL  (Fig.  443),  having  a  diaphragm 
with  a  small  opening  O  between 
them.  It  is  evident  from  the 
symmetry  of  the  combination  that 
any  ray  rr  passing  though  the 
center  of  O  is,  upon  emergence, 
parallel  to  its  incident  direction. 
Therefore  such  rays  cut  the  conjugate  planes  a  and  b  in  simi- 
larly grouped  points.  This  is  strictly  true  if  the  planes  a  and 
b  are  at  equal  distances  from  O  (and  the  magnification  is  —  i). 


Fig.  443. 


LENS    SYSTEMS.  fa 

When  the  magnification  is  other  than  -  i,  the  combination 
remains  sensibly  orthoscopic.  For  photographic  purposes  it  is 
customary  to  use  in  place  of  the  lenses  LL  two  similar  achro- 
matic doublets.  Such  a  combination  is  shown  in  Fig.  454. 

679.  Curvature  of  field. — In  order  to  project  upon  a  screen 
the  most  distinct  image  of  an  extended  flat  object,  which  it  is 
possible  to  form  by  means  of  a  simple  lens,  the  screen  must  be 
curved,  as  shown  by  55  in  Fig.  444.    This  imperfection  of  a  lens 
is    called    curvature    of  field. 

The  curvature  of  field  of  a  lens 
of  given  focal  length  varies 
with  the  refractive  index  of 
the  glass,  and  is  of  opposite 
sign  for  converging  lenses  and 
diverging  lenses.  It  is,  there- 
fore, possible  to  combine  a  converging  lens  and  a  diverging  lens 
of  different  glass,  so  as  to  form  a  doublet  of  any  desired  focal 
length  which  will  give  a  fiat  field.  Flatness  of  field  is  very 
important  in  photographic  objectives,  because  these  are  used  to 
project  images  upon  flat  glass  plates. 

680.  Simultaneous    correction    for    several    imperfections.  - 

There  is  now  a  great  variety  of  optical  glasses  at  the  disposal  of 
the  manufacturing  optician,  and  by  the  combination  of  a  number 
of  lenses,  sometimes  as  many  as  ten,  each  made  of  chosen  glass, 
lens  systems  are  made  which  leave  but  little  to  be  desired.* 
The  apochromatic  microscope  objective  shown  in  Fig.  451, 
which  consists  of  ten  separate  lenses,  is  an  excellent  example 
of  a  combination  designed  to  annul  simultaneously  the  various 
errors  to  which  simple  lenses  are  subject. 

681.  Examples  of  various  lens  systems  used  in  practice. 

(a)  Magnifying  glasses  and  eyepieces.  —  Figures  445-447  show  standard 
forms  of  magnifying  glasses.     Figure  445  shows  the  lens  known  as  Brewster?s 

*  It  may  be  stated  that  wide  field  angle  is  incompatible  with  large  aperture,  and 
that  an  aplanatic  lens  of  large  aperture  cannot  be  orthoscopic. 


62 


ELEMENTS   OF   PHYSICS. 


magnifer;  *  Fig.  446  the  arrangement  called  Wollaston's  doublet.    Figure  447 
is  a  modification  by  Hastings  of  an  earlier  form  by  Brewster.     Figures  448 


Fig.  445. 


FRONT 
Fig.  446. 


Fig.  447. 


and  449  are  achromatic  doublets  such  as  are  used  for  microscope  and  tele- 
scope eyepieces. 

The  lenses  of  Ramsden's  doublet  (Fig.  448)  are  placed  a  little  nearer  together 
than  is  required  (by  equation  330)  to  give  achromatism.     The  result  is  to 


Fig.  448. 


Fig.  449. 


bring  the  focal  points  outside  of  the  combination,  so  that  the  doublet  may  be 
used  to  view  a  real  image,  or  an  object,  just  as  with  an  ordinary  magnifying 
glass. 

The  front  focal  point  t  of  Huygens'  doublet  (Fig.  449)  is  between  the 
lenses,  so  that  this  doublet  cannot  be  used  as  an  ordinary  magnifying  glass. 

The  thing  viewed  with  it  must  be  a  virtual 
image  (as  shown  by  the  arrow  in  Fig.  449)  • 
Telescopes  when  used  for  sighting  are 
always  provided  with  Ramsden's  doublet 
eyepiece  or  some  equivalent  combination. 

(b)  Microscope  objectives.  —  The  apla- 
natic  property  of  the  sphere,  as  described 
in  Art.  657,  is  utilized  in  the  construction 
of  the  microscope  objective  (homogeneous 
immersion  objective)  as  follows  :  A  hemis- 
pherical lens  L  (Fig.  450)  is  mounted  as 
shown.  The  flat  face  of  this  lens  is  sub- 
merged in  oil  of  the  same  refractive  index 

as  the  glass  of  the  lens.  Then  light  from  the  point  P  (of  an  object),  after 
passing  through  the  lens  Z,,  appears  to  have  come  from  a  point  P' .  Addi- 

*  A  simple  lens  is  practically  perfect  when  used  as  a  magnifying   glass    of  low 
power. 

t  This  point  is  the  front  principal  focus.     See  Art.  666. 


Fig.  450 


LENS   SYSTEMS. 


tional  lenses  are  used,  as  shown  by  the  dotted  lines,  to  correct  for  the  chro- 
matic aberration  of  Z,  and  for  concentrating  the  light  from  P'  at  some  point 

beyond,  thus  forming  an  image  of  the  object.     Figure  451 

shows  a  highly  perfected  microscope  objective,  designed  by 

Abbe  and  constructed  by  Zeiss. 

(c}  Photographic  objectives.  —  Figure  452  shows  a  simple 

achromatic  lens  with  diaphragm,  such  as  is  used  for  landscape 

work.    Figure  453  shows  a  standard 

^J'J  DIAPHRAGM 

form  of  wide  angle  orthoscopic  lens 
for  architectural  and  landscape  work.  FR° 
Figure  454  shows  a  standard  form 
of  wide  aperture  orthoscopic  lens. 
Figure  455  shows  a  very  wide  aper- 
ture aplanatic  lens  by  Dalmeyer, 
used  for  portrait  work.  Figure  456 


Fig.  451. 


Fig.  452. 


shows  an  anastigmatic  photographic  lens,  designed  by  P.  Rudolph  and  con- 
structed by  Zeiss  and  by  Bausch  and  Lomb. 


Fig.  453. 


Fig.  454. 


I 

Fig.  455. 


(X)  Telescope  objectives  are  usually  simple  achromatic  doublets.  Such  a 
doublet  may  be  made  both  achromatic  and  aplanatic,  as  explained  in  Art.  696. 
Figure  457  shows  the  standard  form  of  telescope  objective. 


ZEISS  ANASTIGMATIC  SYSTEM 
PHOTOGRAPHIC 
Fig.  456. 


Remark.  —  In  Figs.  445-457  all  the  converging  lenses  are  of  one  or  another 
variety  of  crown  glass,  and  all  the  diverging  lenses  are  of  one  or  another 
variety  of  flint  glass. 


CHAPTER   VI. 
SIMPLE   OPTICAL   INSTRUMENTS. 

682.  Optical   instruments   defined. — The  instruments  to  be 
described  in  this   chapter   are  those   necessary  to  vision  and 
those  used  to  aid  vision  or  to  supplement  it.     Certain  optical 
instruments  which  do  not  come  directly  under  this  definition, 
such  as  the  spectroscope  and  the  polariscope,  will  be  considered 
later. 

683.  The   eye.  —  This  organ  is  shown  in  its  essential  fea- 
tures in   Fig.  458.     The  tough  membrane   of   the  eyeball    is 

sharply  curved  and  transparent  in  front, 
forming  the  cornea  C.  Between  the  cornea 
and  the  crystalline  lens  L  is  a  clear,  watery 
fluid,  the  aqueous  humor.  Behind  the  crys- 
talline lens,  and  filling  the  remainder  of 
the  eyeball,  is  a  clear  semi-fluid  substance, 
Fig.  458.  ^Q  vitreous  humor.  The  front  surface  of 

the  cornea  and  the  surfaces  separating  the  aqueous  humor, 
crystalline  lens,  and  vitreous  humor  are  sensibly  spherical  and 
constitute  a  lens-system  which  projects  an  image  of  external 
objects  upon  a  nervous  membrane,  the  retina,  at  the  back  of 
the  eye.  The  retina  consists  of  a  vast  number  of  minute  end 
organs  of  nerve  fibers  which  come  together,  forming  the  optic 
nerve  O,  and  lead  to  the  brain. 

Two  luminous  points  can  be  seen  as  two  so  long  as  their 
images  on  the  retina  do  not  fall  upon  one  and  the  same  end 

organ.     Over  a  large  portion  of  the  retina  these  end  organs  are 

64 


SIMPLE   OPTICAL   INSTRUMENTS.  65 

relatively  sparse,  but  in  the  thin  portion  /,  called  from  its  color 
the  yellow  spot,  they  are  very  closely  packed  together.  An 
object  can  be  seen  distinctly  only  when  its  image  falls  upon 
this  spot.  The  line  passing  through  the  center  of  the  eye 
lenses  and  the  center  of  the  yellow  spot  is  called  the  axis  of 
the  eye. 

Accommodation.  —  For  distinct  vision,  the  image  on  the  retina 
must  be  sharply  defined.  The  focal  length  of  the  eye  lenses, 
to  give  a  sharply  defined  image  of  an  adjacent  object  upon  the 
retina,  must  be  different  from  their  focal  length  to  give  a  sharp 
image  of  a  distant  object.  This  necessary  variation  of  focal 
length  is  provided  for  by  the  action  of  muscles,  attached  to 
the  edge  of  the  crystalline  lens,  the  contraction  of  which  causes 
the  lens  to  become  thinner  at  the  center.  This  action  is  called 
accommodation.  Ordinarily  the  eye  has  power  of  accommoda- 
tion for  objects  at  any  distance  greater  than  about  15  centi- 
meters from  the  eye.  The  distance  of  most  distinct  vision,  for 
most  individuals,  is  about  25  centimeters. 

The  imperfections  of  the  eye.  —  Some  persons  can  accom- 
modate to  distant  objects  only  with  great  effort,  or  not  at  all; 
such  persons  are  said  to  be  nearsighted.  Persons  who  have 
like  difficulty  in  accommodating  to  near  objects  are  said  to  be 
farsighted.  Nearsightedness  is  relieved  by  the  use  of  spec- 
tacles with  diverging  lenses,  farsightedness  by  the  use  of 
spectacles  with  converging  lenses.  Nearsightedness  (or  far- 
sightedness) may  be  due  to  an  abnormal  elongation  (or  shorten- 
ing) of  the  eyeball  in  the  direction  of  the  axis ;  or  to  an 
abnormally  short  focal  length  (or  long  focal  length)  of  the  eye 
lenses.  Inaccurate  centering  of  the  eye  lenses,  and  more  or 
less  deviation  from  true  spherical  shape  of  the  various  refract- 
ing surfaces,  produce  astigmatism ;  which  is  sometimes  so 
pronounced  as  to  hinder  sharp  vision.  Such  imperfection 
is  corrected  by  the  use  of  spectacles  having  cylindrical  sur- 
faces. 

Apparent  size  of  objects  ;  visual  angle.  —  An  object  appears 


66  ELEMENTS   OF   PHYSICS. 

large  when  its  image  covers  a  large  portion  of  the  retina.  Lines 
drawn  from  the  extremities  of  the  object  through  the  center* 
of  the  eye  lenses,  as  shown  in  Fig.  459,  pass  through  the  ex- 


OBJECT 


Fig.  459. 


tremities  of  the  image.  The  angle,  a,  between  these  lines  is 
called  the  visual  angle  of  the  object  and  is  a  measure  of  its 
apparent  size. 

684.  The  photographic  camera  consists  of  a  light-tight  box, 
in  the  front  of  which  a  lens  is  mounted.  At  the  back  of  this 
box  is  a  sensitive  plate,  upon  which  an  image  of  external  objects 
is  projected  by  means  of  the  lens.  The  requirements  of  photo- 
graphic work  have  led  to  the  construction  of  a  variety  of  lens 
systems,  designed  to  give  rapidity  of  action,  wideness  of  angle, 
flatness  of  field,  exquisiteness  of  definition,  etc.  Special  lenses 
are  made  for  use  in  portraiture,  in  architectural  work,  in  land- 
scape, in  instantaneous  photography,  etc.  Some  of  these  forms 
are  described  in  Chapter  V.  The  color  correction  of  photo- 
graphic lenses  is  made  with  reference  to  those  wave  lengths 
by  which  the  sensitive  plates  are  most  strongly  affected. 

685;  The  magic  lantern  is  an  arrangement  for  projecting  the 
image  of  a  brilliantly  illuminated  object  or  picture  upon  a 
screen.  The  light  from  a  brilliant  lamp  L  (Fig.  460)  passes 
through  condensing  lenses  CC,  through  a  transparent  picture 
55,  and  through  an  objective  lens  O,  which  throws  a  greatly 

*  Strictly,  lines  pass  through  the  extremities  of  the  image,  which  are  drawn 
through  the  posterior  nodal  point  of  the  lens  system  of  the  eye,  parallel  respectively 
to  lines  drawn  from  the  extremities  of  the  object  to  the  anterior  nodal  point.  (Com- 
pare Art.  670.) 


SIMPLE   OPTICAL   INSTRUMENTS. 


67 


enlarged  image  of  .S.S  upon  the  distant  screen.  Symmetrical 
orthoscopic  combinations,  such  as  are  designed  for  photographic 
work,  are  much  used  in  magic  lantern  objectives.  Such  an 
objective  is  shown  in  Fig.  460. 


a     0 


c     c 


Fig.  460. 

686.  The  simple  microscope ;  definition  of  magnifying  power. 

—  The  simple  microscope,  or  magnifying  glass,  consists  of  a 
converging  lens  (simple  or  compound)  which  is  held  near  the 
eye.  The  object  to  be  examined  is  moved  up  until  it  is  seen 
sharply.  When  this  is  the  case,  the  eye  is  looking  at  a  virtual 
image  of  the  object,  and  this  virtual  image  is  at  the  distance  of 
most  distinct  vision  from  the  eye.  The  magnifying  power  of 
a  microscope  is  denned  as  the  ratio  of  the  apparent  size  of  an 
object,  as  seen  with  the  microscope,  divided  by  its  apparent  size, 
as  seen  with  the  naked  eye  at  a  distance  of  25  centimeters. 

687.  Proposition.  —  The  magnifying  power  of  a  magnifying 
glass  is 

•»=^+'.'  (33D 

in  which/  is  the  principal  focal  length  of  the  lens  in  centimeters. 
The  eye  is  supposed  to  be  accommodated  for  a  distance  of  25 
centimeters. 

Proof.  —  Let  O  (Fig.  461)  be  the  object,  and  i  its  virtual 
image,  formed  by  the  magnifying  glass.  The  eye  being  accom- 
modated to  25  centimeters,  the  object  will  be  moved  until  the 


68 


ELEMENTS   OF   PHYSICS. 


image  i  is  25  centimeters  from  the  eye.  The  distance  from 
the  lens  to  the  eye  may  be  neglected,  so  that  the  distance  b 
of  the  image  from  the  lens  may  be  taken  as  25  centimeters. 


Since  the  distance  from  a  lens  to  a  virtual  image  is  considered 
negative,  we  have,  from  equation  (327), 


Whence 


25  +/ 

Now  the  visual  angle  a  is  sensibly  the  same  as  the  angle  at 
c  subtended  by  O  and  it  and  this  angle  is  as  many  times  as  great 
as  the  angle  that  O  would  subtend  at  a  distance . of  '25  centimeters, 
as  25  is  greater  than  a :  that  is, 

m=—-  (iii) 

a 

Substituting  the  value  of  a  from  (ii)  in  (iii),  we  have  equation 

(331).  Q.E.D. 

Remark  i.  — When  the  eye  is  accommodated  for  parallel  rays, 

b  (Fig.  461)  is  infinity,  and  a  —p.     So  that  m  —  — • 

Remark  2. — A  magnifying  power  of  125  diameters  (focal 
length  of  2  millimeters)  is  about  as  great  as  can  be  obtained 
satisfactorily  with  the  simple  microscope.  For  greater  magni- 
fying powers  the  compound  microscope  is  more  satisfactory. 

688.  The  compound  microscope  consists  of  a  lens  A  (Fig.  462) 
which  gives  an  enlarged  real  image  i  of  an  object  ot  and  a 


SIMPLE   OPTICAL   INSTRUMENTS. 


69 


magnifying  glass  B  for  viewing  this  image.  The  lens  A  is 
called  the  objective.  It  is  usually  a  lens  system  of  the  form 
shown  in  Fig.  450,  or  in  Fig. 

Q 

451.     The  lens  B  is  called  the 

eyepiece;    it    likewise    usually 

consists  of  a  system  of  lenses 

as  shown  in  Fig.  448,  or  Fig. 

449.      Figure   463   shows,  full 

size,  the  actual  arrangement  of  the  essential  optical  parts  of  a 

compound  microscope.*     The  group  of  lenses  C  is  a  condenser 

for  illuminating  the  object  which  is  placed  upon  the  stage  of 

the  instrument ;    O  is  a  homogeneous  immersion  objective,  2 

millimeters  focal  length,  by  Bausch  and  Lomb,  and  E  is  a  Huy- 

gens  doublet  eyepiece. 


Fig.  463. 

The   magnifying  power  of  a  compound 


689.   Proposition. 

microscope  is 


in  which  /  is  the  principal  focal  length  of  the  eye  lens  (or 
system),  and  a  and  b  are  the  respective  distances  of  the  object 
and  image  from  the  center  of  the  objective  lens,  as  shown  in 


*  For  more  detailed  information  concerning  microscopes  see  Gage,  The  Micro- 
scope, 5th  ed.,  1896.     Andrus  and  Church,  Ithaca,  N.Y. 


70  ELEMENTS   OF   PHYSICS. 

Fig.  462.  If  the  objective  is  a  lens  system,  then  a  and  b  are 
the  distances  of  object  and  image  from  the  respective  nodal 
points  of  the  system. 

Proof.  —  The  image  is  -  times  as  large  as  the  object,  and  the 
eyepiece,  being  a  magnifying  glass,  makes  this  image  appear 

2  % 

—  +  I  times  as  large  as  it  would  appear  to  the  naked  eye  at  a 

distance  of  25  centimeters,  or-f  —  +  I J  times  as  large  as  the 

object  would  appear  to  the  naked  eye  at  that  distance.  Q.E.D. 
By  the  use  of  high  grade  objective  systems,  such  as  have  been 
described  in  Chapter  V.,  and  of  a  doublet  eyepiece  of  the  form 
shown  in  Fig.  449,  satisfactory  definition  and  sufficient  bright- 
ness can  be  obtained  with  a  magnifying  power  as  high  as  1 500 
diameters. 

690.   The  telescope  consists  of  a  lens  of  long  focus  O  (Fig.  464), 
which  forms  an  image  i  of  a  distant  object ;  and  of  a  magnifying 


Fig.  464. 

glass  E  for  viewing  this  image.  The  lens  O  is  called  the  object 
glass,  and  is  usually  a  system  of  lenses  of  the  kind  shown  in 
Fig.  457-  The  lens  E  is  called  the  eyepiece,  and  it  is  usually 
a  Ramsden  or  Huygens  doublet  (Figs.  448  and  449). 

One  of  the  most  important  features  of  an  astronomical  tele- 
scope is  light-gathering  power,  which  requires  an  objective  lens 
of  large  aperture.  Until  very  recently  the  difficulties  of  mak- 
ing large  lenses  were  so  great  that  concave  mirrors  were  used 
instead  in  all  the  largest  telescopes.  This  method  of  construe- 


SIMPLE   OPTICAL   INSTRUMENTS.  7! 

tion  had  its  culmination  in  the  great  reflecting  telescope  of 
Lord  Rosse  (1842),  the  aperture  of  which  was  about  183  centi- 
meters. The  development  of  the  art  of  glass  making  is  now 
such  that  it  is  possible  to  construct  nearly  perfect  lenses  with 
an  aperture  of  100  centimeters.  The  performance  of  a  large 
refracting  telescope  is  greatly  superior  to  that  of  the  reflecting 
telescope,  and  the  latter  has  become  obsolete. 

691.  The  magnifying  power  of  a  telescope  is  defined  as  the 
quotient  of  the  visual  angle  «  (Fig.  464)  of  the  object  as  seen 
with  the  telescope,  divided  by  the  visual  angle  ft  of  the  object 
as  seen  with  the  naked  eye.  (Compare  Art.  686.)  If  m  be  the 
magnifying  power,  we  have  therefore 


If  we  consider  the  object  to  be,  sensibly,  at  an  infinite  dis- 
tance, iO  in  Fig.  464  is  the  principal  focal  length  /  of  the 
object  glass.  If  the  eye  is  accommodated  for  parallel  rays, 
the  distance  iE  is  the  principal  focal  length  of  the  eyepiece. 
The  angles  «  and  ft  are  then  proportional  to  /  and  p1  ,  so  that 


From  (i)  and  (ii)  we  have, 


=  (333) 


The  magnifying  power  of  a  telescope  is  therefore  equal  to  the 
ratio  of  the  focal  length  of  the  object  glass  to  the  focal  length  of 
the  eyepiece. 

692.  The  use  of  the  telescope  for  sighting.  —  The  telescope 
attached  to  such  instruments  as  transits  and  levels,  and  to  a 
great  variety  of  apparatus  in  physical  laboratories,  is  used  for 
sighting.  A  cross  of  very  fine  wires  is  fixed  in  the  focal  plane 
of  the  object  glass  so  as  to  be  seen  through  the  eyepiece 
(Ramsden's  doublet)  at  the  same  time  with  the  image  of  a  dis- 
tant object.  When  the  image  of  a  distant  point,  as  a  star,  falls 


ELEMENTS   OF   PHYSICS. 


upon  the  point  of  intersection  of  these  wires,  the  axis  of  the 
telescope  points  exactly  at  the  distant  point.  The  axis  of  the 
telescope  is  the  line  drawn  from  the  intersection  of  the  cross 
wires  to  the  center  of  the  object  glass. 

693.   The  erecting  telescope  or  spyglass.  —  The  simple  tele- 
scope shows  objects  inverted.    The  spyglass  is  a  telescope  modi- 

0 


Fig.  465.  — Diag.  of   spyglass. 

fied  so  as  to  make  distant  objects  appear  erect.  Figure  465 
shows  the  arrangement  of  parts  in  a  spyglass.  The  arrow  i 
represents  an  inverted  image  of  a  distant  object  formed  by  the 
object  glass  O.  A  lens  vS  forms  at  i1  an  inverted  image  of  i. 
This  image  i'  is  therefore  erect,  and  is  viewed  by  an  eye  lens  E 
as  before.  In  practice  the  lenses  O  and  E  are  lens  systems. 
The  lens  5  is  usually  a  symmetrical  doublet  somewhat  similar 
to  Fig.  443.  The  introduction  cf  the  erecting  system  5 
lengthens  the  telescope  very  materially. 

694.   The  opera  glass  is  a  telescope  provided  with  a  diverging 
lens  as  an  eyepiece.      The   action   is    shown    in    Fig.  466,  in 

which  i  is  the  position  which 
the  (inverted)  image  formed 
by  the  object  glass  O  would 
take  were  it  not  for  the  di- 
verging lens  E.  The  action 
of  the  lens  E  is  to  form  at 
i'  an  enlarged  inverted  virtual 
image  of  i.  In  looking  in 
through  E,  the  observer  sees  the  erect  image  i'  of  the  distant 
object.  This  telescope  is  very  short  for  a  given  magnifying 
power. 


Fig.  466. 


CHAPTER   VII. 


DISPERSION. 

695.  Newton's  experiment.  Homogeneous  light;  non-homo- 
geneous light.  —  A  beam  of  parallel  rays  of  white  light,  such  as 
sunlight  or  lamplight  B  (Fig.  467),  is  changed  into  a  fanlike 
beam  B'  by  a  prism.  This  fanlike  beam  falling  upon  a  screen 
55  produces  an  illuminated  band  R  V,  called  a  spectrum,  which 
is  red  at  the  end  R  and  passes  by 
insensible  gradations  through 
orange,  yellow,  green,  and  blue 
to  violet  at  the  end  V.  The 
beam  of  light  B  is  said  to  be 
dispersed  by  the  prism.  The 
fanlike  beam  B'  produces  white 
illumination  when  concentrated 
by  a  converging  lens  upon  a 
small  portion  of  a  screen. 

A  photographic  plate  reveals  the  existence  of  invisible  rays 
beyond  V,  the  ultra-violet  rays,  especially  in  sunlight;  and  a 
thermopile  or  bolometer  shows  the  existence  of  rays  inside  of 
or  below  R,  the  infra-red  rays.  The  portion  of  the  spectrum 
between  R  and  V  is  called  the  visible  spectrum. 

A  narrow  beam  B"  passing  through  a  small  hole  in  the 
screen  is  deflected  by  a  prism,  but  not  dispersed.  The  action 
of  the  prism  upon  the  beam  of  white  light  shows  that  white 
light  is  non-homogeneous,  being  made  up  of  dissimilar  parts. 
The  beam  B" ,  on  the  other  hand,  is  homogeneous.  Homogene- 
ous light  is  sometimes  "called  monochromatic  light. 

73 


Fig.  467. 


74 


ELEMENTS   OF    PHYSICS. 


Since  the  prism  P  (Fig.  467)  deflects  each  homogeneous 
beam  B"  differently,  it  is  obvious  that  the  glass  of  which  it  is 
made  has  different  refractive  indices  for  the  various  homogene- 
ous components  of  white  light. 

The  phenomena  of  interference  (see  Chapter  VIII.)  show  that 
a  homogeneous  beam  of  light  is  a  simple  wave  train  of  definite 
wave  length,  or,  more  strictly  speaking,  of  definite  frequency. 
A  composite  beam,  such  as  a  beam  of  sunlight,  consists  of  a 
group,  sometimes  infinite  in  number,  of  such  wave  trains. 
After  dispersion,  each  simple  wave  train  assumes  a  different 
path  according  to  its  wave  length. 

696.  The  achromatic  lens. — A  prism  of  flint  glass  which 
gives  a  spectrum  of  the  same  length  (say  from  red  to  blue) 
as  a  prism  of  crown  glass,  produces,  on  the  whole,  much  less 
deflection  of  the  beam  than  does  the  crown-glass  prism. 

If,  therefore,  the  two  prisms  are  placed  together  as  in  Fig. 
468,  the  dispersion  of  the  one  will  be  counterbalanced  by 
that  of  the  other,  but  the  deflection  will  not  be  wholly  an- 
nulled. The  compound 

FLINT  .  ^,  f  .,,       . 

prism  therefore  will  give 
deflection  without  dis- 
persion. 

Spectra  of  the  same 
total  length,  produced 
by  a  flint  prism  and 
by  a  crown  prism  (Fig.  474),  are  not  of  the  same  length 
from  red  to^yellow,  yellow  to  green,  etc.  ;  so  that  a  flint  prism 
cannot  be  made  to  wholly  neutralize  the  dispersion  of  a  crown- 
glass  prism.  A  prism  of  crown  glass  which  gives  the  same 
mean  deflection  as  a  prism  of  flint  glass  gives  a  much  broader 
spectrum  than  the  latter,  so  that  two  such  prisms  may  be 
arranged  to  give  a  very  good  spectrum,  the  middle  part  of  which 
is  not  deflected.  Such  an  arrangement  of  prisms  is  used  in  the 
direct  vision  spectroscope.  The  action  is  rendered  more  satis- 


DISPERSION. 


75 


factory  by  using  two  crown   prisms.     Figure  469  shows   the 
action  of  such  a  compound  prism. 

A  compound  lens,  to  be  achromatic,  must  satisfy  the  condi- 
tion expressed  by  equation  (334),  which  is  derived  as  follows  : 


/r~i 
/  \ 


Fig.  469. 

Let  fj/  and  p"  be  the  refractive  indices  of  crown  glass  for  red 
and  blue  respectively,  and  '  nj  and  //,/'  the  corresponding  refrac- 
tive indices  of  flint  glass.  Let  C  (Fig.  470)  be  a  converging 
crown-glass  lens  and  F  a  diverging  flint-glass  lens. 
Let  *  h  be  the  thickness  of  the  crown  lens  at  the  center 
and  h'  the  thickness  of  the  flint  lens  at  the  edge,  and 
let  d  be  the  diameter  of  the  lenses.  Consider  the 
retardation  of  the  central  portion,  relative  to  the  edge 
portion,  of  a  plane  wave  of  red  light.  The  retardation 
by  the  crown  lens  is  (//  —  i)/z,  and  the  retardation  by 
the  flint  lens  is  —  (/*/  —  i)h!  (see  Art.  660);  the  total 
retardation  is  therefore  (//  —  i)h  —  (/*/  —  i)/i'.  Simi- 
larly,  the  total  retardation  of  the  central  portion  of  a 
plane  wave  of  blue  light  is  (//'  —  i)  h  —  (^  —  i)  h1  .  If  these 
two  waves  are  to  be  focused  at  the  same  point,  the  two  retarda- 
tions must  be  equal,  which  gives  at  once 


F 


This  is  the  necessary  relation  between  h  and  h1  to  give  achro- 
matism. The  absolute  values  given  to  h  and  h'  depend  upon 
the  diameter  of  the  lens  and  the  desired  focal  length.  The 


*  The   crown   lens  is  assumed  to  come  to  a  sharp  edge  and  the  flint  lens   is 
assumed  to  be  of  zero  thickness  at  the  center  for  the  sake  of  simplicity. 


76  ELEMENTS   OF   PHYSICS. 

curvatures  of  the  various  surfaces,  in  so  far  as  they  are  not 
fixed  by  the  absolute  values  of  h  and  h1 ',  are  chosen  so  as  to 
make  the  system  aplanatic. 

697.  The  spectroscope.  —  In  the  spectrum  obtained  by  New- 
ton (described  in  Art.  695)  the  beam  which  falls  upon  the  prism 
is  admitted  through  a  circular  hole.  Each  beam  of  homogeneous 
light  is  as  wide,  however,  as  the  incident  beam ;  so  that  the 
various  homogeneous  beams  overlap  greatly.  This  difficulty  is 
overcome  by  means  of  the  spectroscope.  In  this  instrument 
(Fig.  471)  the  light  to  be  analyzed  is  passed  through  a  very 
narrow  slit,  S,  between  two  straight  metal  edges.  This  slit, 


-V 


Fig.  471, 


which  may  be  regarded  as  the  source  of  the  light,  is  at  the  prin- 
cipal focus  of  an  achromatic  lens  L,  called  the  collimating  lens. 
The  spherical  waves  from  the  various  points  of  the  slit  are  plane 
after  passing  through  L.  These  plane*  waves  pass  through  the 
prism  P,  and  each  homogeneous  component  of  the  light  (i.e.  each 
simple  wave  train)  appears  to  have  come  from  a  distinct  slit,  and 
the  lens  L'  forms  images  of  the  slit  side  by  side  at  R  V,  one  for 
each  homogeneous  component.  This  band  of  images  is  called  a 
spectrum  ;  it  is  viewed  by  a  magnifying  glass  or  eyepiece  E. 
The  images  of  the  slit  are  called  the  lines  of  the  spectrum. 

*  Spherical   waves   are   much    distorted   when  refracted  at  a   plane  surface,  as 
described  in  Art.  652;    hence  the  importance  of  the  collimating  lens  L. 


DISPERSION. 


77 


698.  Continuous   spectra.  —  The  light   from   hot   solids   and 
liquids  has  in  it  wave  trains  of  every  wave  length,  and  when 
analyzed  by  the  spectroscope  the  images  of   the   slit   form    a 
continuous  band,  red  at  one  end  and  violet  at  the  other. 

Well-known  examples  of  continuous  spectra  are  the  spectra 
of  the  candle  flame,  the  petroleum  flame,  the  gas  flame,  etc. 
The  light  from  such  flames  is  given  off  by  particles  of  carbon. 
The  spectra  of  the  lime  light,  of  the  magnesium  flame,  of  the 
incandescent  lamp,  and  of  the  crater  of  the  electric  arc  are 
likewise  continuous. 

699.  Bright-line  spectra.  —  The  light   from  a  hot  vapor  or 
gas  contains,  ordinarily,  only  wave  trains  of  certain  definite  wave 
lengths,  and  gives,  when  analyzed  by  the  spectroscope,  a  group 
of   distinctly  separated  images  of   the  slit  (bright  lines)  with 
intervening  dark  spaces.      Such  spectra  are  called  bright-line 
spectra.      Every  gas  or  vapor  has  a  characteristic   spectrum, 
that  is,  a  characteristic  grouping  of  images  of  the  slit  (bright 
lines)  at  R  V. 

The  bright-line  spectra  of  some  of  the  metals  are  of  very 
simple  character.  When  a  salt  of  lithium,  for  example,  is  vapor- 
ized in  the  flame  of  a  Bunsen  burner,  a  single  red  line  appears 
in  the  field  of  the  spectroscope.  Sodium  is  characterized  by 
the  presence  of  two  yellow  lines.  These  are  so  near  together 
that  unless  the  slit  is  narrow,  the  dispersion  large,  and  the  defi- 
nition good,  they  appear  as  a  single  line.  Thallium  has  a  spec- 
trum which,  as  ordinarily  observed,  possesses  a  single  line  of 
brilliant  green.  The  spectrum  of  indium  in  like  manner  ex- 
hibits a  single  blue  line.  When  carefully  studied  under  condi- 
tions of  more  intense  incandescence,  these  elements  are  found 
to  have  more  complicated  spectra. 

Sodium  vapor,  in  the  electric  arc,  shows  some  sixteen  lines  in 
the  visible  spectrum ;  thallium,  twenty-five  lines.  Even  in  the 
infra-red  of  the  spectrum  narrow  regions  of  great  intensity 
have  been  discovered  by  means  of  the  bolometer.  Thus 


ELEMENTS   OF  PHYSICS. 


Snow  *  found  two  strong  lines  due  to  sodium  which  are  quite 

invisible   to   the   eye.      These   are   shown  in  Fig.  472,  which 

1000 

,              shows  the  distribution  of  energy  in  the  spectrum 

950 

of  that  metal.     The  region  between  HH  and  A,  in 

900 

that  diagram  comprises  the  entire  visible  spectrum. 

850 

- 

All  lying  to  the  right  of  A  constitutes  the  infra-red. 

800 

• 

The  most  complex  bright-line  spectrum  is  prob- 

750 

• 

ably  that  of  iron.     Thousands  of  lines 

700 
650 

due  to  the  vapor  of  that  metal  have 

600 

. 

already  been  identified  and  listed,  and 

3    550 
i- 

- 

the  catalogue  is  doubtless  incomplete. 

§    500 

- 

z    450 

- 

700.   Dark-line  spectra.  —  When  an 

400 

• 

intense  beam  of  light  passes  through 

350 
300 

a  vapor  or  a  gas,  those  wave  trains  are 

250 

absorbed  which  the  gas  itself  gives  off. 

200 

_ 

(Kirchhoff   and   Bunsen.)     Such  light 

150 

- 

in    the     spectroscope    gives     missing 

100 

- 

images  of  the  slit  or  dark  lines  where 

50 
1 

•jLU 

LA 

\L~S~~ 

bright  lines   would 

W-  su^*i_WVW.     i 

f     ,,0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2.0               OCCUr     in     tflC     SpCC- 

trum  of  the  gas  or 
Fig'472-  vapor.      The   most 

striking  example  of  dark-line  spectra  is  the  solar  spectrum, 
which  shows  a  great  number  of  dark  lines.  These  dark  lines 
in  the  solar  spectrum  were  first  carefully  studied  by  Fraunhofer 
(1819) ;  they  are  called  Fraunhofer  s  lines.  The  more  promi- 
nent of  these  lines  are  designated  by  the  letters  A,  B,  C,  D,  E, 
b,  F,  G,  Hp  and  H2,  beginning  with  the  red  end  of  the  spec- 
trum. Figure  473  shows  the  visible  portion  of  the  sun's 
spectrum,  as  it  appears  when  dispersed  with  a  single  prism. 
The  figure  is  from  a  drawing  by  Becquerel.  The  significance 
of  the  Fraunhofer  lines  was  first  pointed  out  by  Kirchhoff 
and  Bunsen  in  1865,  as  stated  above.  They  found  groups  of 

*  Physical  Review,  Vol.  I.,  p.  95. 


DISPERSION. 


79 


dark  lines  in  the  sun's  spectrum  to  coincide  with  the  groups 
of  bright  lines  given  by  iron  vapor,  sodium  vapor,  hydrogen, 
and  other  substances,  and  they  inferred  the  existence  of  rela- 
tively cool  masses  of  these  vapors  in  the  sun's  atmosphere. 


Kirchhoff   and  Bunsen    devised  the  following  experiment   for 
showing  the  significance  of  dark-line  spectra : 

Using  the  flame  of  a  Bunsen  burner,  supplied  with  sodium 
vapor  by  the  vaporization  of  common  salt,  they  obtained  the 
ordinary  bright-line  spectrum  of  sodium.  Then,  passing ,  an 
intense  beam  of  light  from  a  lime  light  through  the  Bunsen 
flame  into  the  slit,  it  was  found  that  the  absorption  of  the 
sodium  vapor  was  such  as  to  leave  relatively  dark  lines  in  place 
of  the  bright  lines  given  by  the  flame  alone.  This  experiment 
has  become  classical  under  the  name  of  the  reversal  of  sodium 
lines. 

701.  The  spectrometer.  —  This  is  a  spectroscope  which  is 
provided  with  a  divided  circle  by  means  of  which  the  lines  of 
the  spectrum  may  be  definitely  located.  The  essential  parts  of 
a  spectrometer  are  shown  in  Fig.  471.  The  lenses  L'  and  E  are 
respectively  the  objective  and  eyepiece  of  a  telescope,  which 
turns  about  a  pivot  at  the  center  of  the  circle  and  is  provided 
with  a  cross  wire  in  the  focal  plane  RV. 

Consider  a  beam  of  homogeneous  light  which,  upon  emer- 
gence from  the  prism,  is  focused  by  the  lens  L' .  When  the 
telescope  is  set  so  that  its  cross  wire  coincides  with  the  line 
of  the  spectrum,  which  is  due  to  the  homogeneous  beam  in 
question,  then  the  axis  of  the  telescope  is  parallel  to  the  beam, 


BRA 

OF  THE 

.  ERSITY 


8o  ELEMENTS   OF   PHYSICS. 

and  the  reading  upon  the  divided  circle  of  a  vernier,  which 
moves  with  the  telescope,  determines  the  location  of  the  spec- 
tral line. 

Determination  of  the  refractive  index.  —  The  total  angle  of 
deflection  of  a  given  homogeneous  beam  may  also  be  determined 
by  the  process  just  described.  When  this  (minimum)  angle  of 
deflection  has  been  measured,  and  the  angle  of  the  prism  is 
known,  the  refractive  index  of  the  prism  is  easily  calculated. 

i 

702.  Normal  and  prismatic  spectra.  —  The  dispersion  by  glass 
prisms  is  such  that  the  distribution  of  the  images  of  the  slit  in 
the  continuous  spectrum  is  by  no  means  uniform.  Towards  the 
red  there  is  relative  crowding  together,  while  towards  the  violet 
end  of  the  spectrum  the  dispersion  increases  rapidly.  A  spec- 
trum in  which  the  distribution  is  uniform,  i.e.  in  which  equal 
distances  in  the  spectrum  correspond  everywhere  to  equal  dif- 
ferences of  wave  length,  is  called  a  normal  spectrum.  The 
relation  of  the  normal  spectrum  to  prismatic  spectra  produced 
by  prisms  of  flint  glass  and  of  crown  glass,  is  indicated  in 
Fig.  474. 

NORMAL  SCALE 
4000  5000  6000  7000 


I  I  I          I         I       I      1      !     I     I        I       Mill 

4000  5000  6000  7000 

PRISMATIC  SCALE  (FLINT) 

Fig.  474. 

703.  The  spectrophotometer.  —  This  instrument  is  a  spectro- 
scope arranged  for  determining  the  composition  *  of  light.  The 
measurements  consist  in  comparing  the  intensity  of  each  part  of 
the  spectrum  of  a  given  source  of  light  with  the  intensity  at  the 
same  part  of  the  spectrum  of  a  source  which  has  been  selected 
as  a  standard.  Figure  475  shows  a  convenient  form  of  this  in- 

*  That  is,  the  relative  intensity  of  the  various  homogeneous  components. 


DISPERSION. 


81 


strument.  It  consists  of  a  direct  vision  spectroscope  (Art.  696) 
mounted  upon  a  carriage  which  travels  along  a  track  between 
the  two  sources,  the  spectra 
of  which  are  to  be  compared. 
The  slit  of  the  instrument,  5 
(Fig.  476),  is  horizontal.  Two 
prisms,  /,  /,  reflect  (totally) 
the  light  from  a  standard  lamp 
and  from  the  source  of  light 
L,  which  is  to  be  compared 
with  the  standard,  into  the 
two  ends  of  the  slit.  The 
two  spectra  are  seen  side  by  side  in  the  field  of  the  spectro- 
scope. The  attention  is  fixed  upon  a  narrow  portion  of  the 
two  spectra  (say  the  red),  and  the  instrument  is  moved  along 
the  line  IF  until  this  narrow  portion  is  of  equal  brightness 


Fig.  475. 


STANDARD  LIGHT 


~T ® 


Fig.  476. 


in  the  two  spectra.  From  the  distances  /  and  /'  the  relative 
intensities  of  the  sources,  for  the  given  portion  of  the  spec- 
trum, can  then  be  computed.  This  process  is  repeated  until 
the  entire  spectrum  has  been  explored.  (See  further  the 
chapter  on  Photometry.) 

There  are  many  other  devices  by  means  of  which  the  two 
spectra  to  be  compared  in  spectrophotometry  may  be  reduced 
to  equal  brightness.  Sometimes  the  widths  of  the  two  ends  of 
the  slit  are  varied ;  sometimes  a  wedge  of  neutrally  tinted  glass 
is  interposed  in  the  path  of  the  brighter  beam  ;  sometimes  pairs 
of  Nicol  prisms  (see  the  chapter  on  Polarization)  are  used. 


CHAPTER   VIII. 
INTERFERENCE   AND   DIFFRACTION. 

704.  Interference  from  similar  sources.  —  Consider  two  points, 
O  and  Of  (Fig.  477),  which  send  out  continuously  simple  wave 
trains  of  wave  length  X.  The  points  O  and  O1  may  be  thought 
of  as  luminous  points,  as  periodic  disturbances  on  the  surface 
of  water,  or  as  two  tuning  forks  in  unison  sending  out  sound 


Fig.  477. 

waves.  Consider  a  point  q  whose  distance  from  O  is  equal  to 
its  distance  from  O',  or  differs  from  it  by  a  whole  number  of 
wave  lengths  n\.  The  wave  trains  from  O  and  O'  are  continually 
alike  in  phase  at  such  a  point,  and  they  work  together  to  pro- 
duce disturbance  there.  On  the  other  hand,  at  a  point/,  whose 

distance  from  O  differs  from  its  distance  from  O1  by  an  odd 

82 


INTERFERENCE   AND   DIFFRACTION.  83 

number  of  half  wave  lengths,  the  wave  trains  are  continually 
opposite  in  phase  and  tend  to  annul  each  other. 

All  the  points  q  which  satisfy  the  above  condition  lie  on  the 
dotted  hyperbolas  *  (Fig.  477)  of  which  O  and  O'  are  the  foci. 
These  dotted  hyperbolas  intersect  the  line  OO'  at  equal  intervals 
x(JX).  One  is  a  straight  line  bisecting  OO'. 

All  the  points/  which  satisfy  the  above  condition  lie  on  the 
full-line  hyperbolas  midway  between  the  dotted  hyperbolas. 

Along  the  dotted  lines  of  Fig.  477  the  disturbance  due  to  the 
two  sources  O  and  O'  is  great,  and  along  the  full  lines  the  dis- 
turbance is  small  or  in  some  cases  zero.  If  the  light  from  O 
and  O'  falls  on  a  screen  AB,  bright  bands  of  illumination  will 
be  produced  where  AB  intersects  the  dotted  hyperbolas  (hyper- 
boloids  of  revolution  about  OO'  as  an  axis),  and  dark  bands  will 
be  left  along  the  intersections  of  AB  with  the  full-line  hyper- 
boloids. 

This  phenomenon  is  called  interference.  The  light  and  dark 
bands  on  the  screen  are  called  interference  fringes ;  two  such 
sources  as  O  and  O'  are  called  similar  sources. 

705.  Displacement  of  fringes.  —  If  O  and  O'  (Fig.  477)  are  two 
tuning  forks  not  exactly  in  unison ;  then  as  one  of  them,  say  O, 
gains  on  the  other,  the  hyperboloids  will  all  move  downwards 
(in  the  figure),  and  when  half  a  complete  vibration  has  been 
gained,  the  /  and  q  hyperbolas  will  have  changed  places.  Thus 
at  any  stationary  point  strong  and  weak  sound  will  alternate, 
producing  what  are  called  beats.  All  the  methods  for  showing 
interference  of  light  depend  upon  the  production  of  identically 
similar  sources,  and  the  phenomenon  of  moving  fringes,  which 
are  most  striking  in  the  case  of  sound,  cannot  be  observed  in  that 
of  light.  If,  however,  a  slip  of  thin  glass  G  (Fig.  477)  be  placed 
between  O'  and  the  screen,  the  advancing  wave  train  will  be 
retarded  in  passing  through  the  glass,  thus  falling  behind  its 

*  The  distances  from  a  point  on  an  hyperbola  to  the  two  foci  have  a  constant 
difference. 


84 


ELEMENTS   OF   PHYSICS. 


former  place,  and  the  fringes  will  be  displaced  along  the  screen 
from  A  towards  B. 


706.   Distribution  of  fringes  over  a  screen.  —  Consider  two 


similar  sources  O,  O'  (Fig.  478). 


Fig.  478. 


Let  2  a  be  the  distance  between 
them,  and  b  the  distance  of  a 
screen  AB.  Let  d  and  d'  be 
the  distances  of  O  and  O1 
x  from  a  point  p  of  the  screen 
distant  x  from  the  point  C. 

When     a'-d  =  —,          (i) 

2 

and  ^  is  an  even  number,  the 
'     point  /  is  at  the  center  of  a 
is  at   the   center  of  a  dark 


bright   band.     When  n  is  odd, 
band. 

Now  d'  —  d  =  OJm,  and,  since  the  triangles  OO'm  and 

.,,..,  ,          O'm     x 

sensibly  similar,  we  have  — —  =  -,  or 


OO b 


(ii) 


n\ 


Writing  —  for  d'  —  d  and  2  a  for  OO1,  and  solving  for  x, 


we  have 


nb\ 


(335) 


which  expresses  the  distances  x  from  the  center  of  the  screen 
to  the  various  fringes.  For  light,  X  is  very  small  and  the  ratio 

-  must  be  very  large  if  the  successive  fringes  are  to  be  far 

enough  apart  to  be  distinguishable.  Equation  (335)  enables  the 
calculation  of  X  when  x,  a,  and  b  have  been  observed.  The 
number  n  may  be  easily  counted,  being  two  for  the  first  bright 
band,  four  for  the  next,  and  so  on.  It  is  found  in  this  way  that 
for  the  extreme  violet  light  of  the  spectrum  the  value  of  X  is 


INTERFERENCE  AND   DIFFRACTION.  85 

about  39-icr6  centimeters,  and  for  the  extreme  red  light  about 
75.icr6  centimeters. 

707.  Colored  fringes.  —  If  the  similar  sources  O,  O'  (Figs.  477 
and  478)  give  off  white  light  instead  of  homogeneous  light,  then 
a  set  of  fringes  will  be  produced  by  each  simple  wave  train,  and 
the  effect  on  the  screen  AB  will  be  the  superposition  of  all  the 
fringes  thus  produced.     The  central  fringe  will  be  white,  for 
this  is  a  bright  fringe  for  every  wave  length.     Passing  out  from 
this  center,  fringes  take  on  a  succession  of  colors,  due  to  the 
extinction,  one  after  another,  of  violet,  blue,  green,  yellow,  and 
red. 

708.  Arrangements  for  producing  interference  fringes. 

(a)  Newton  s  arrangement.  —  Newton  who  was  among  the 
first  to  observe  interference  phenomena,  employed  two  narrow 
slits,  close  together,  as  described  in  Art.  714. 


\ 

\ 
\ 

°\  \  x/ 

\     A  • 

V      \  D 

\ 

S 

Fig.  479. 

(b)  FresneVs  mirrors.  —  Light  from  a  very  small  source  S 
(Fig.  479)  falls  upon  two  mirrors  M,  M',  as  shown.  After  re- 
flection from  M  and  Mf,  the  direction  of  the  rays  is  such  that 
the  light  seems  to  come  from  the  points  O  and  O1 ',  and  inter- 
ference fringes  are  produced  on  the  screen  AB. 


86  ELEMENTS    OF   PHYSICS. 

(c)  Lloyd's  mirror.  —  A  screen  AB  (Fig.  480)  is  illuminated  di- 
rectly by  a  small  source  O,  also  by  the  light  from  O  which  strikes 
the  mirror  and  is  reflected  to  the  screen. 
The  latter  appears  to  have  come  from 
O',  thus  producing  interference  fringes 
on  the  screen. 

f  i        MIRROR  (d}   Sound  fringes.  —  Using  a   shrill 

Fig  480  whistle   giving    sound   wave    trains    of 

about     2\    centimeters     wave    length, 

Stevens  and  Mayer  have  obtained  marked  interference  effects. 
They  used  arrangements  similar  to  Fresnel's  mirrors  and  to 
Lloyd's  mirror.  The  regions  of  intense 
disturbance  were  indicated  by  means  of 
a  sensitive  flame.  Such  a  flame  is  de- 
picted in  Fig.  481.  When  undisturbed, 
the  flame  is  long  and  narrow  (b),  and 
when  disturbed,  it  changes  to  the  shorter 
form  (a)  shown  in  the  figure. 

709.    The  colors  of  thin  plates. —The 

interference  effects  above  described  seldom 
or  never  occur  to  ordinary  observation. 
Thin  plates,  on  the  other  hand,  present 
very  striking  interference  phenomena, 
which  are  of  common  occurrence.  The 
colors  of  soap  films,  and  of  films  of  oil  on 
water,  are  familiar  examples.  The  action 
of  a  thin  film  in  producing  interference 

Fig.  481.  •      i      •    n  r    n 

is  briefly  as  follows  : 

Consider  a  simple  train  of  waves  T  (Fig.  482)  of  wave  length 
A,,  incident  upon  a  thin  transparent  plate  PP.  A  portion  T1  of 
this  train  is  reflected  from  the  surface  A  with  change  of  phase 
(see  Art.  623).  The  remainder  of  the  train,  passing  on  through 
the  plate,  reaches  the  second  surface  B,  and  is  partly  reflected 
(without  change  of  phase).  This  portion,  after  passing  back 


INTERFERENCE   AND   DIFFRACTION.  87 

through  the  plate,  emerges  (in  part)  into  the  air,  and  travels 
as  the  train   T1  parallel  to  the  train  T.     When  the  distance 
2  a  is  an  odd  number  of   half  wave  lengths,  then,   since  the 
reflection  at  the  surface  A  is  with 
change  of  phase,  the  two  reflected 
trains  T'  and  T"  are  in  like  phase, 
and   give   a   resultant    train    of    in- 
creased   intensity.     When    the   dis- 
tance 2  a  is  an  even  number  of  half 


wave    lengths,    the    two     reflected  B 

trains    T  and    T'  are  in  opposite 

phase,  and  tend  to  annul  each  other.  If  the  incident  light  T 
is  non-homogeneous  (white  light),  all  those  homogeneous  com- 
ponents of  the  white  light  are  greatly  strengthened  whose  half 
wave  lengths  are  contained  in  the  distance  2  a  an  odd  number 
of  times,  and  those  homogeneous  components  are  greatly  weak- 
ened whose  half  wave  lengths  are  contained  in  the  distance  2  a 
an  even  number  of  times. 

If  the  plate  is  very  thin,  i.e.  if  a  is  small,  then,  of  the  various 
visible  homogeneous  components  of  white  light  (X=/5  x  icr6 
centimeters  for  red,  to  X  =  39.  io~6  centimeters  for  violet)  only 
one  or  two  will  satisfy  the  above  conditions,  and  the  plate  will 
appear  brilliantly  colored.  If  the  plate  is  thick,  however,  then 
a  great  number  of  the  components  will  satisfy  the  condition. 
In  this  case  the  strengthened  components  and  the  weakened 
components  will  be  distributed  more  or  less  evenly  throughout 
the  spectrum,  and  the  plate  will  not  show  perceptible  color. 

If  the  reflected  light  T1  +  T" ,  in  this  latter  case,  is  analyzed 
by  the  spectroscope,  the  spectrum  will  be  crossed  by  numerous 
dark  bands  corresponding  to  the  wave  lengths  which  are  weak- 
ened, showing  that  in  case  of  thick  plates  the  interference 
takes  place,  although  no  color  is  to  be  perceived  by  the  eye. 
The  value  of  a  (Fig.  482)  depends  evidently  upon  the  obliquity 
of  the  incident  wave  train,  as  well  as  upon  the  thickness  of  the 
plate. 


88  ELEMENTS   OF   PHYSICS. 

710.  Newton's  rings. — The   thin   film   of   air   between  two 
glass  plates  laid  together  presents  a  fine  show  of  interference 
colors.     Newton  made  a  very  minute  study  of  this  effect,  using 
the  air  film  between  a  flat  glass  plate  and  the  convex  surface  of 
a  lens  laid  upon  it.     In  this  case  the  film  increases  in  thickness 
from  the  point  of  contact  outwards.     With  homogeneous  light 
a  succession  of  light  and  dark  rings  surrounds  this  point.    With 
white  light  these  rings  are  colored.* 

711.  Diffraction.  —  The  spreading  of  a  wave  disturbance  into 
the  region  behind  an  obstacle  is  called  diffraction.     This  action 
has  been  discussed  in  Chapter  II.  to  the  extent  of  determining 
the  degree  of  approximation  to  which  a  wave  disturbance  may 
be  considered  to  be  propagated  in  straight  lines,  and  the  size 
an  object  must  have  in  order  that  it  may  completely  screen  off 
a  disturbance  from  a  point  behind  it.     We  shall  now  consider 
what  takes  place  in  the  region  throughout  which  a  wave  dis- 
turbance does  spread  sensibly.     The  diffraction  of  plane  waves 
past  straight  edges  is  the  simplest  case,  and  the  discussion  here 
given  is  limited  to  that  case.     The  diffraction  of  spherical  waves 
and  diffraction  past  curved  edges  gives  rise  to  phenomena  which 
are  essentially  similar  to  the  above. 

712.  Half-period  zones  with  reference  to  a  line.  —  Let  TT 

(Fig.  483)  be  a  train  of  plane  waves  of  wave  length  X  approach- 
ing a  screen  55.  Let  O  be  a  line  perpendicular  to  the  paper 
along  which  the  illumination  produced  by  TT  is  to  be  con- 
sidered, with  a  view  to  the  determination  of  the  effect  of  an 
obstacle  in  the  fixed  plane  AB,  distant  b  from  O.  With  O  as  an 
axis,  describe  circular  cylinders  (not  shown  in  the  diagram),  one 
of  which  has  a  radius  b,  the  next  a  radius  b  -f-  -,  the  next  a 
radius  b  +  X,  the  next  a  radius  b  +  3 —  and  so  on.  These 

2 

cylinders  will  cut  the  plane  AB  into  bands  or  zones  parallel 

*  See  Preston,  Theory  of  Light,  pp.  114-168. 


INTERFERENCE  AND   DIFFRACTION. 


89 


to  O.  Calling  the  middle  zone  No.  I,  the  distance  from  P 
to  the  inner  edge  of  the  nth  zone  is  ±  ^/ nb\,  as  explained  in 
Art.  629.  The  middle  zone  is  the  broadest ;  the  successive 
zones  grow  narrower  above  and  below  P  and  approach  the 
limiting  width  — 


1 

1 

1 

1 

|  s 

1  ' 
1 

1 

1 

I 
1 
1 

1  ^ 

1  > 

1 

1 
1 

O 


B 


Fig.  483. 


For  the  purposejof  the  following  discussion  it  is  sufficient  to 
know  that  the  portion  of  the  disturbance  along  O  which  is  con- 
tributed by  the  nth  zone  is,  on  the  whole,  opposite  in  phase  to  that 
contributed  by  the  n  +  I  th  zone,  and  that  this  contribution  grows 
less  and  less  as  n  increases. 

The  illumination  along  O  produced  by  the  unobstructed  train 
of  waves  is  called  normal  illumination. 

713.   Diffraction  past  the  straight  edge  of  a  large  obstacle. 

(a)  For  points  outside  of  the  geometrical  shadow.  —  When  the 
edge  of  the  obstacle  (which  is  parallel  to  O)  is  at  P,  as  shown 
by  the  dotted  line  in  Fig.  483,  half  of  the  middle  zone  is  cut  off, 
together  with  all  the  zones  below  P,  and  the  illumination  along 
O  is  one-quarter*  normal.  If  the  obstacle  is  moved  downwards, 
the  middle  zone  will  gradually  be  uncovered  and  the  illumination 


*  The  amplitude  of  the  disturbance  at  O  is  one-half  normal,  and  therefore  (Art. 
620)  the  intensity  is  one-quarter  normal. 


9o 


ELEMENTS    OF   PHYSICS. 


along  O  will  increase,  reaching  a  maximum  considerably  greater 
than  normal  at  about  the  time*  when  that  zone  is  wholly 
uncovered.  As  the  next  zone  (No.  2)  becomes  uncovered,  the 
illumination  at  O  falls  off,  because  the  contribution  from  this 
zone  is  nearly  opposite  in  phase  to  the  contribution  from  the 
middle  zone,  and  reaches  a  minimum  considerably  less  than 
normal  at  about  the  time  when  the  second  zone  is  wholly  un- 
covered. As  the  third  zone  becomes  uncovered,  the  illumina- 
tion along  O  reaches  a  maximum,  and  so  on.  The  maxima  and 
minima  grow  less  and  less  pronounced  as  the  obstacle  moves 
»  down,  and  the  illumination  along  O  becomes  sensibly  constant 
and  equal  to  normal  when  the  obstacle  has  receded  so  far  as 
to  uncover  as  many  as  six  or  eight  zones  below  P. 

(b)  For  points  within  the  geometrical  shadow.  —  Let  the  un- 
obstructed portion  of  AB  (Fig.  484)  be  broken  up  into  half- 


r 

c 

i 

1 

i  * 
i 

i 

i     P 

• 

I 

i  . 

i  -> 

OBSTACLE 

i 

i 

i 

Fig.  484. 

period  zones  beginning  at  the  edge  of  the  obstacle.  In  this  case 
the  band  next  the  edge  is  more  effective  in  producing  disturb- 
ance at  O  than  the  next  band,  the  effect  of  which  at  O  is  opposite 
in  phase,  and  so  on.  There  is,  therefore,  a  slight  resultant 
disturbance,  or  illumination,  at  O  produced  by  all  the  bands 
above  the  edge  of  the  obstacle.  This  resultant  illumination 
has  a  quarter  of  its  normal  value  when  the  edge  is  opposite  O. 

*  Strictly,  when  0.87  of  the  lower  half  of  the  middle  zone  is  uncovered. 


INTERFERENCE   AND   DIFFRACTION. 


As  the  obstacle  moves  upward  towards  A,  the  illumination  at 
O  falls  off  continuously. 

The  variation  of  illumination  at  O,  as  the  obstacle  moves,  is 
the  same  as  the  variation  of  illumination  along  55  with  a  given 
position  of  the  obstacle.  The  ordinates  of  the  curve  in  figure 
485  represent  the  intensities  of  illumination  along  a  screen,  as 
described  above  under  (a)  and  (£). 


GEOMETRICAL 
SHADOW. 

DISTANCE  OF  OBSTACLE 
94  C.  M. 


HORIZONTAL  DIMENSIONS 
ARE  AMPLIFIED  10  TIMES. 


Fig.  485. 

714.  Diffraction  through  a  slit.  —  Consider  the  illumination 
reaching  the  point  O  (Fig.  486)  through  the  slit  W.  When  the 
slit  is  far  above  or  below  P,  the  illumination  at  O  is  sensibly 
zero.  As  the  slit  moves  up  along  AB,  the  half-wave  zones  in 
the  slit  become  broader  and  the  number  of  them  in  the  slit 
grows  less.  When  there  is  an  even  number  of  zones  in  the 
slit,  they  neutralize  each  others'  action  at  O  and  the  illumination 
is  zero.  When  there  is  an  odd  number  of  zones  in  the  slit,  the 


^2  ELEMENTS    OF   PHYSICS. 

effect  of  one  of  them  is  left  outstanding  and  the  illumination  at 
O  is  a  maximum.  The  intensities  of  these  maxima  increase 
rapidly  as  the  slit  approaches  P. 


w 


Fig.  486. 

Remark.  —  The  light  which  passes  through  a  slit,  not  more 
than  half  a  wave  length  broad,  passes  out  in  all  directions  from 
the  slit  very  much  as  from  a  row  of  luminous  points.  (See  Fig. 
487.)  Two  such  slits  near  together  act*  as  similar  sources  and 


B 

Fig.  487. 

produce  interference  fringes  on  a  screen  SS,  as  explained  in 
Art.   704. 

The   regions  of  maximum  and  minimum  disturbance  (the  / 
and  q  regions  described  in  Art.  704)  produced  by  the  sound 

*  Provided  the  light  T  has  come  from  a  very  small  source. 


INTERFERENCE   AND   DIFFRACTION. 


93 


coming  from  a  shrill  whistle  through  two  adjacent  narrow  slits 
in  a  wall  have  been  traced  by  Mayer  and  Stevens,  and  by 
Rayleigh,  who  used  the  sensitive  flame  as  an  indicator,  as  ex- 
plained in  Art.  708. 

715.   Diffraction  past  the  edges  of  a  narrow  strip.  —  Consider 
an  obstacle  OO'  (Fig.  488).     The  illumination  on  the  screen 


1 

1 

1 

S> 

1 

0 

i 

OSSTACL: 

ft 

1 

1 
1 

0' 

1 

1 

1 

1 
1 

3 

Fig.  488. 

outside  the  geometrical  shadow  varies  according  to  a  law  which 
is  nearly  the  same  as  that  which  applies  when  the  obstacle  is 
infinitely  broad.  Inside  the  geometrical  shadow  ab,  the  light 
comes  from  regions  very  near  the  edges  O  and  Of,  so  that  O 
and  O1  act  as  similar  sources  and  produce  interference  fringes 
similar  to  those  described  in  Art.  704. 

One  of  the  few  cases  in  which  the  phenomena  of  the  diffrac- 
tion of  light  occur  to  ordinary  observation  is  that  of  the  shadows 
of  wires  and  twigs  cast  upon  frosted  windows  by  a  distant  arc 
light. 

716.  Zone  plates.  —  A  transparent  plate  having  opaque  bands 
painted  upon  it,  so  that  when  placed  in  the  position  of  AB 
(Fig.  483)  every  alternate  half-period  zone  is  obscured,  is  called 
a  zone  plate.  With  such  a  plate  the  disturbance  from  each 
unobscured  zone  reaches  O  in  the  same  phase,  and  the  resul- 


94 


ELEMENTS   OF   PHYSICS. 


tant  illumination  at  O  is  very  intense.  The  action  of  a  zone 
plate  is  shown  by  Fig.  489  in  which  AB  is  a  plate  on  which 
the  middle  zone  (No.  i)  and  the  odd  zones  above"  and  below 
are  obscured  as  shown. 

Consider  the  wavelet  starting  out  from  the  center  of  the 
second  zone  when  a  given  wave  of  the  train  TT  reaches  AB, 
the  wavelet  which  started  out  from  the  center  of  the  fourth 
zone  when  the  first  preceding  wave  of  TT  reached  AB,  the 


Fig.  489. 

wavelet  which  started  out  from  the  center  of  the  sixth  zone 
when  the  second  preceding  wave  of  TT  reached  AB,  and  so  on. 
These  wavelets  are  tangent  to  the  circle  (cylinder)  CC  with  its 
center  at  O,  and  consequently  they  work  together  to  produce  a 
wave  front  converging  upon  O,  and  the  disturbance  at  O  is  very 
intense. 

717.  The  diffraction  grating  consists  of  a  set  of  a  large  num- 
ber of  equidistant  slits  in  an  opaque  screen.  The  distance 
between  the  slits  is  called  the  grating  space.  Let  TT  (Fig.  490) 


INTERFERENCE   AND   DIFFRACTION. 


95 


be  an  incident  simple  train  of  plane  waves  of  wave  length  X, 
parallel  to  a  diffraction  grating  AB,  of  which  the  grating  space 
is  d.  Cylindrical  wavelets  pass  out  from  the  slits  as  the  suc- 
cessive waves  of  the  incident  train  reach  AB.  Consider  the 


Fig.  490. 


nth  wavelet  from  the  slit  Slt  the  2  nth  wavelet  from  the  slit 
S2,  the  3  nth  wavelet  from  the  slit  53,  and  so  on.  These  wave- 
lets are  tangent  to  the  plane  CD,  making  an  angle  0  with  AB 
such  that 


.    n      n\ 

sin  6  =  —r. 
a 


(336) 


These  wavelets  work  together  to  produce  a  plane  wave  CD. 
The  (n  —  i)th  wavelet  from  slit  Sv  the  (2n—  i)th  wavelet 
from  slit  52,  the  (3/2—  i)th  wavelet  from  slit  Sz,  etc.,  work 
together  to  produce  another  plane  wave  parallel  to  CD,  and  at 
a  distance  X  behind  it,  and  so  on.  Therefore  there  is  a  train  of 
plane  waves  of  wave  length  X,  parallel  to  CD,  passing  out  from 
the  grating.  If  the  incident  light  TT  is  non-homogeneous,  as, 
for  example,  sunlight,  as  many  distinct  wave  trains  will  be  pro- 
duced on  either  side  of  the  normal  to  AB  as  there  are  simple 
wave  trains  in  the  incident  light.  If  a  lens  L'  be  placed  in  the 
path  of  these,  each  diffracted  wave  train  will  be  brought  to 
focus  at  a  distinct  point  O. 


96 


ELEMENTS   OF   PHYSICS. 


The  points  O,  Or,  O",  etc.,  each  illuminated  by  one  homogene- 
ous component  of  the  incident  light,  constitute  what  is  called  the 
diffraction  spectrum.  The  two  spectra,  one  on  either  side  of 
the  normal  to  AB,  produced  when  n  =  I,  are  called  spectra  of 
the  first  order ;  the  two  spectra  produced  when  n  =  2  are  called 
spectra  of  the  second  order,  etc.  In  general,  all  these  spectra 
are  produced  simultaneously,  and  the  group  of  points  O,  O1  O", 
etc.,  which  constitutes  one  spectrum,  often  overlaps  upon  the 
group  which  constitutes  the  spectrum  of  the  next  higher  order. 
The  complete  action  of  the  diffraction  grating  is  very  strik- 
ingly shown  by  the  wire  grating  sometimes  used  in  front  of 
the  object  glass  of  an  astronomical  telescope.  This  wire  grat- 
ing consists  of  a  large  number  of  parallel  and  equidistant  wires 
stretched  in  a  frame.  The  object  glass  then  produces  a  central 
white  image  of  the  star.  On  either  side  of  this  are  the  groups 
of  images  constituting  the  spectra  of  the  first  order,  beyond 
these  the  groups  of  images  constituting  the  spectra  of  the 
second  order,  and  so  on.  A  grating  which  has  a  large  number 
of  accurately  spaced  slits  produces  very  pure  spectra,  that  is, 
each  point  O  (Fig.  490)  is  illuminated  by  only  one  homogeneous 
component  of  the  incident  light.  When  the  slits  are  not  very 

narrow,  compared  with  the  grating 
space,  then  some  of  the  spectra 
of  higher  orders  are  weakened. 
For  example,  when  the  width  of 
the  slits  is  -,  all  spectra  of  even 
orders  are  missing.* 

718.    Action  of  a  diffraction  grat- 
ing  upon   obliquely   incident   plane 
waves.  —  Consider  a  train  of  plane 
Fis- 491-  waves     TT    (Fig.    491),    of    wave 


*  For  a  more  complete  discussion  of  the  diffraction  grating,  see  Preston's  Light, 
pp.  186-201. 


INTERFERENCE   AND   DIFFRACTION. 


97 


length  X,  approaching  a  grating  AB.  Consider  a  certain 
wave  W  of  the  incident  train.  Let  the  wavelets  which  pass 
out  from  the  various  slits  as  this  wave  W  reaches  the  slits 
be  designated  as  wavelets  No.  i  ;  let  the  wavelets  which  passed 
out  as  the  preceding  wave  passed  the  slits  be  designated  as 
wavelets  No.  2,  etc.  The  wave  W  having  just  reached  the  slit 
55  has  yet  a  distance  $x  to  go  before  reaching  the  fifth  slit 
below  55,  that  is,  the  slit  S ;  so  that  the  wavelets  which  pass 
out  from  S5  are  $x  ahead  of  the  wavelets  which  pass  out  from 
the  slit  5.  Consider  the  nth  wavelet  from  the  slit  S,  the  2  nth 
wavelet  from  the  slit  S%,  the  3  nth  wavelet  from  the  slit  53,  etc. 
These  wavelets  are  tangent  to  CD  (Fig.  491)  ;  therefore,  since 

the  distance  SS5  is  5  d,  we  have  sin  0  =  — — ^—  ',  and  since, 

y 

moreover,  sin  i  =  — ,  we  have 
a 

sin0  =  ^  +  sinf.  (337) 

a 

719.  Reflection  gratings  and  transmission  gratings.  —  Diffrac- 
tion gratings  having  small  grating  space  are  made  by  ruling 
equidistant  lines  on  glass  or  speculum  metal.     The  former  is 
called  a  transmission  grating;  the  action  of  such  a  grating  is 
discussed  in  the  foregoing  articles.     The  latter  is  called  a  reflec- 
tion grating.     When  light  is  reflected  from  such  a  grating,  it 
virtually  comes  through  the  spaces  between  the  rulings,  so  that 
the  action  of  the  reflection  grating  is  similar  to  the  action  of 
the  transmission  grating. 

720.  The  grating  spectrometer ;  measurement  of  wave  lengths. 
A    reflection    grating    AB    (Fig.    492)    is    mounted    upon    an 
arm  R  which  turns  about  a  pivot  at  the  center  of   a  divided 
circle  by  means  of  which  the  grating  may  be  turned  through 
any  measured  angle.     The  light  to  be  analyzed   is    admitted 
through  a  narrow  slit  S,  which  is  at  the  principal  focus  of  a 
lens  L.     After  passing  this  lens,  the  light  becomes  an  incident 
train  of  plane  waves   TT.     The  various  diffracted  wave  trains 

H 


98 


ELEMENTS   OF   PHYSICS. 


are  concentrated  at  the  points  O,  O",  in  the  focal  plane  of  the 
lens  L'  and  viewed  by  the  eye  lens  E.     To  determine  the  wave 

length  of  a  given  diffracted  wave 
train,  proceed  as  follows  :  Turn 
the  arm  R  until  the  unscratched 
portions  of  the  grating  surface 
reflect  light  from  the  slit  directly 
back  to  the  slit  again,  and  read 
the  vernier  V.  Let  this  reading 
be  r.  Turn  the  grating  until 
light  from  the  cross  wire  O'  is 
reflected  directly  back  to  the 
cross  wire  again,  and  read  the 
vernier.  Let  this  reading  be  rf. 
Then  having  turned  the  arm  R 
until  the  light  from  vS  is  reflected  into  the  telescope,*  turn  it 
until  the  desired  diffracted  train  CD  is  concentrated  at  O' 
(the  first  time  this  occurs  will  be  for  the  spectrum  of  the 
first  order,  the  second  time  will  be  for  the  spectrum  of  the 
second  order,  etc.  Having  turned  to  the  spectrum  of  the  third 
order,  say,  the  value  of  n,  equation  (337),  will  be  three)  and 
read  the  vernier  a  third  time.  Let  this  reading  be  rff.  Then 
the  angle  i  is  equal  to  r—r",  and  the  angle  6  is  equal  to 
r"  —  r1 .  The  wave  length  X  of  the  train  CD  may  be  calcu- 
lated from  equation  (337)  when  the  grating  space  is  known. 

The  following  table  gives  the  wave  lengths  for  the  principal 
Fraunhofer  lines  as  measured  by  Bell : 

TABLE. 


Fig.  492. 


Line. 


A 7594.06  X  I0~8  cm. 

B 6867.46 

C 6563.06 

Di {5896-15 

D2 15890.18  " 

EI {527°-5° 

E2 (.5269.72          " 

*  The  lenses  L'  and  E  constitute  a  telescope. 


Line. 


F 4861.49 

G  (4308.07 

'  (.4307.90 

HI 4101.85 

H2 3968.62 


INTERFERENCE   AND   DIFFRACTION. 


99 


The  most  complete  and  accurate  table  of  wave  lengths  is  that 
published  by  H.  A.  Rowland  in  the  Astro-Physical  Journal 
(1895-96). 

Wave  length  units.  —  It  is  often  convenient  to  express  wave 
lengths  in  terms  of  some  small  fraction  of  a  centimeter. 

The  micron  (symbol  /i)  is  one  millionth  of  a  meter  or  to 
io~4  cm. 

o  o 

The  Angstrom  unit  (symbol  A)  is  io~8  cm. 

721.  The  concave  grating.  —  Consider  the  distances  a  and  b 
(Fig.  493)  of  two  points  6*  and  S'  from  a  point  /  upon  a  surface 
AB.  As  the  point  p  moves  along  AB, 
the  sum  a  +  b  changes  continuously.  If 
only  those  portions  of  the  surface  AB  are 
polished  (the  intervening  portions  being 
roughened  by  scratching)  for  which  p\ 

where  n  is  an  integer  and  X  the  wave 
length  of  homogeneous  light  coming  from 
S,  then  all  the  wavelets  from  these  polished 
portions  will  be  in  like  phase  at  S',  producing  intense  illumina- 
tion—  producing,  in  fact,  an  image  of  5  at  S'.  Rowland  has 
realized  the  condition  expressed  in  equation  (i)  in  his  concave 
grating.  AB  (Fig.  493)  is  a  speculum  metal  surface,  with  its 


Fig.  493. 


I 


Fig.  494. 


center  of  curvature  at  C,  and  is  ruled  with  lines  which  are 
the  intersections  with  AB  of  equidistant  parallel  planes.  The 
dotted  line  is  a  circle  upon  Cp  as  a  diameter.  The  grating  AB 
forms  as  many  distinct  images  of  5  along  the  dotted  circle  near 


I0o  ELEMENTS   OF   PHYSICS. 

Sf  as  there  are  distinct  homogeneous  components  in  the  light 
coming  from  5.  Any  light  to  be  analyzed  may  be  sent  through 
a  narrow  slit  at  S,  and  the  group  of  images  at  S'  may  be 
observed  direct  or  allowed  to  fall  upon  a  sensitive  plate  and 
thus  photographed.  Figure  494  is  a  copy  of  a  photograph,  taken 
in  this  way,  of  a  portion  of  the  solar  spectrum  in  the  neighbor- 
hood of  Fraunhofer's  lines  Dl  and  D2. 


CHAPTER   IX. 


COLOR. 

722.  Sensations  of  brightness  and  color.  —  The  special  sensa- 
tions due  to  light  are  brightness  and  color.     Brightness  depends, 
for  a  given  homogeneous  light  stimulus,  upon  the  amplitude  of 
vibration ;  that  is  to  say,  upon  the  intensity  of  the  stimulus. 
The  various  sensations  of  color  are  due  to  differences  of  wave 
length,  when  the  light  which  produces  the  sensation  is  homo- 
geneous, or  to  differences  in  the  composition  when  the  light  is 
mixed. 

723.  Luminosity  of  homogeneous  light. — The  intensity  of  the 
sensation  produced  by  a  beam  of  light  is  called  its  luminosity 
or  brightness.     The  luminosity  of  the  various  portions  of  the 
visible  spectrum  differs  greatly. 


80 


GO 


40 


20 


4500 

BLUE 


5000     5500    6000    6500 
Fig.  495. 


7000 

RED 


If,  for  example,  we  hold  a  printed  page  in  the  different  regions 
of  the  spectrum  and  vary  the  brightness  of  the  source  of  light, 

inr 


IO2  ELEMENTS   OF   PHYSICS. 

from  which  the  spectrum  is  obtained,  until  it  is  just  possible 
to  read  the  text,  we  find  the  necessary  brightness  to  be  much 
greater  in  the  red  or  violet  than  in  the  yellow  or  green.  The 
luminosity  of  one  region  of  the  spectrum  is  to  the  luminosity 
of  another  region  as  the  necessary  brightness  of  the  source  in 
the  second  case  is  to  the  necessary  brightness  of  the  source  in 
the  first  case.  The  curve  given  in  Fig.  495  was  obtained  in  this 
way.  The  ordinates  of  this  curve  represent  the  luminosities  of 
the  various  portions  of  the  spectrum  of  gaslight. 

Remark.  —  The  luminosity  of  the  various  regions  of  the  spec- 
trum is  not  at  all  proportional  to  the  energy  intensity  of  those 
regions.  The  energy  intensity  of  the  spectrum  of  gaslight  is 
a  maximum  in  the  infra-red,  where  the  luminosity  is  of  course 
zero,  and  falls  off  rapidly  towards  the  violet. 

724.  Colors  due  to  homogeneous  light.  —  The  various  wave 
lengths  of  homogeneous  light  produce  a  variety  of  color  sen- 
sations.     Newton  recognized  and  named  seven  colors  in  the 
spectrum  :  red,  orange,  yellow,  green,  blue,  indigo,  and  violet. 
About  one  hundred  and  fifty  steps  are  made,  however,  in  going 
through  the  spectrum  from  one  tint  to  the  next  which  can 
barely  be  distinguished   from    it.      That  is  to  say,  there  are 
about  one  hundred  and  fifty  distinguishable  tints  in  the  spec- 
trum.    Various   compound   names   have   come   into   use   such 
as  orange-yellow,  bluish-green,  violet-blue,  etc.,  and   these  are 
applied  to  the  regions  between  those  named  by  Newton. 

725.  Colors  due  to  mixed  light.     Definition  of  white  light.  — 
Sunlight,  or  any  light  approaching  it  in  composition,  is  called 
white  light.     The  sensation  produced  by  such  light  (aside  from 
complications  growing  out  of  contrast  effects)  is  called  white. 
Mixed  light  produces,  in  general,  deeper  and  deeper  color  sen- 
sations as  it  deviates  more  and  more  in  composition  from  white 
light.     Colored  bodies  occurring  in  nature  owe  their  color  to 
the  fact  that  they  send  to  the  eye  light  of  which  the  composi- 


COLOR. 


103 


tion  differs  more  or  less  widely  from  the  composition  of  white 
light. 

Color  by  selective  radiation.  —  Hot  gases  give  off  light  which 
differs  widely  from  white  light  in  composition.  Thus,  most  of 
the  color  effects  in  fireworks  are  produced  by  the  use  of  salts  of 
various  metals,  such  as  strontium  and  copper.  These  salts  are 
vaporized  and  the  hot  vapors  give  off  brilliantly  colored  light. 

Color  effects  by  selective  reflection  and  selective  transmission.  — 
Many  substances  such  as  pigments  and  dyestuffs  reflect  or 
transmit  in  excess  certain  wave  lengths  of  the  light  which  falls 
upon  them.  Such  substances  appear  colored  when  illuminated 
by  white  light.  When  such  substances  are  illuminated  by  yel- 
lowish gaslight,  their  colors  become  less  Brilliant,  and  when  illu- 
minated by  homogeneous  light,  as  for  example  sodium  light, 
the  differences  between  their  colors  disappear  entirely. 

Colors  by  interference.  —  The  light  reflected  from  a  thin  plate? 
a  soap  film  for  example,  may  have  one  or  more  of  its  homo- 
geneous components  extin- 
guished and  others  strengthened 
by  interference.  Such  light  dif- 
fers widely  from  white  light  in 
composition,  and  produces  bril- 
liant color  sensations.  The 
colors  produced  by  diffraction 
are  essentially  interference  ef- 
fects. 

The  action  of  bodies  in  modi- 
fying the  composition  of  light 
by  reflection  and  transmission 
is  discussed  at  length  in  Chapter 
XII. 


5000  6000 

Fig.  496. 


7000 


726. ,  The  specification  of  the 
composition  of  light ;  exam- 
ples. —  Choose  any  light,  say  gaslight,  as  a  standard  of  com- 


104 


ELEMENTS   OF   PHYSICS. 


position.  The  intensity  at  each  part  of  the  spectrum  of  this 
light  is  to  be  taken  as  the  unit  intensity  for  that  part  of  the 
spectrum.  The  composition  of  any  other  light  is  then  easily 
specified  by  giving  the  intensity  or  brightness  of  each  part  of 
its  spectrum  in  terms  of  the  intensity  of  the  same  part  of  the 


POTASSIUM  CHROMATE 


ULTRAMARINE    BLUE 


4000 
VIOLET 


5000 


7000 
RED 


4000 
VIOLET 


Fig.  497. 


5000 


Fig.  498. 


7000 
RED 


spectrum  of  the  standard  light,  as  determined  by  means  of  the 
spectrophotometer.  The  curves  in  Fig.  496  show  the  composi- 
tion of  daylight,  of  the  light  from  the  glow  lamp,  and  of  lime 
light,  each  referred  to  gaslight.  The  curve  in  Fig.  497  shows 
the  composition  of  gaslight  after  passing  through  a  solution  of 
potassium  chromate  (referred  to  gaslight  direct).  The  curve 
in  Fig.  498  shows  the  composition  of  gaslight  reflected  from 
ultramarine  blue  (referred  to  gaslight  direct). 

727.  Color  mixing.  Dichroic  and  Trichroic  Vision.  —  Consider 
two  (or  more)  beams  of  light  of  different  composition.  These 
beams  give  distinctly  different  sensations  of  color.  If  they 
are  mixed  (i.e.  allowed  to  enter  the  eye  and  fall  upon  the  same 


COLOR.  I05 

portion  of  the  retina),  the  sensation  is  still  that  of  a  single  defi- 
nite color.  The  various  beams  of  light  are  said  to  blend. 

Any  color  may  be  matched  by  properly  mixing  a  deep,  or 
saturated,  red  light,  a  saturated  green  light,  and  a  saturated 
violet  light.  To  this  end,  the  intensity  of  each  colored  light  must 
be  under  control  and  varied  tintil  the  mixture  matches  the  given 
color.  This  is  an  experimental  fact  first  pointed  out  by  Thomas 
Young. 

For  some  persons  (red  blind)  any  color  may  be  matched  by 
properly  mixing  a  saturated  green  light  and  a  saturated  violet 
light.  For  other  persons  (green  blind)  any  color  may  be  matched 
by  properly  mixing  a  saturated  red  light  and  a  saturated  violet 
light. 

Persons  for  whom  any  color  may  be  matched  by  mixtures  of 
two  saturated  colors  are  said  to  possess  dichroic  vision.  Normal 
vision,  or  the  vision  of  the  majority,  is  called  trichroic  vision. 
Dichroic  vision  is  sometimes  called  color  blindness.  The  color 
sense  of  a  person  having  dichroic  vision  is  strikingly  different 
from  the  color  sense  of  a  person  having  trichroic  vision.  (See 
Arts.  729  and  730.)  About  four  per  cent  of  the  male  population 
and  about  four  out  of  every  thousand  of  the  female  population 
of  the  civilized  world  possess  dichroic  vision. 

The  color  top. — The  most  convenient  arrangement  for  mixing 
colored  light  is  by  means  of  the  color  top.  This  consists  of  a 
rotating  spindle  which  carries  three  colored  circular  disks  (red, 
green,  and  blue)  slitted  in  such  a  way  that  any  desired  sector  of 
the  face  of  each  disk  may  be  exposed  to  view.  When  this  triple 
disk  is  rotated  rapidly,  the  colors  blend  and  give  a  single  sensa- 
tion of  color,  the  tint  of  which  may  be  modified  at  will  by 
varying  the  amounts  of  exposure  of  the  respective  disks. 

728.  The  Young-Helmholtz  theory  of  color.  —  The  experimen- 
tal fact  stated  in  the  previous  article  led  Thomas  Young  (1801) 
to  infer  the  existence  of  three  primary  color  sensations. 

Helmholtz  attributed  each  of  these  primary  sensations  to  a 


I06  ELEMENTS   OF   PHYSICS. 

distinct  set  of  nerves  in  the  retina  of  the  eye.  The  nerves 
which  upon  excitation  give  the  primary  sensation  of  red  are 
called  the  red  nerves ;  those  which  upon  excitation  give  the 
primary  sensation  of  green  are  called  the  green  nerves ;  and 
the  set  which  upon  excitation  gives  the  primary  sensation 
of  violet,  the  violet  nerves.  Simultaneous  excitation  of  all  three 
sets  of  nerves  gives  a  blended  sensation,  the  character  of  which 
depends  upon  the  degrees  of  excitation  of  the  respective  sets 
of  nerves. 

The  number  of  distinct  color  sensations  produced  by  the 
action  in  varying  proportions  of  the  countless  homogeneous 
rays  of  which  the  visible  spectrum  is  composed  is  very  large 
(according  to  Titchener,*  about  30,000,  and  according  to  Rood  f 
a  much  larger  number). 

A  person  having  dichroic  vision  has  only  two  primary  color 
sensations  and  only  two  sets  of  color  nerves.  Persons  who  do 
not  have  the  primary  sensation  of  red  are  said  to  be  red  blind, 
persons  who  do  not  have  the  primary  sensation  of  green  are 
said  to  be  green  blind. 

The  Young-Helmholtz  theory  of  color  is  not  the  only  theory 
in  vogue  ;  indeed,  physiologists  are  inclined  to  reject  it  for 
lack  of  microscopical  evidence  of  the  existence  of  the  three 
sets  of  nerves;  and  psychologists  are  inclined  to  reject  it, 
mainly,  because  of  the  difficulty  of  explaining  the  great  number 
of  distinguishable  qualities  of  color  sensation  by  the  varying 
intensity  or  quantity  of  sensation  on  three  sets  of  nerves.  The 
theory  however  gives  a  very  clear  representation  of  the  experi- 
mental facts  stated  in  the  previous  article,  and  a  very  satis- 
factory explanation  of  contrast  effects  and  of  color  blindness. 

Intensity  of  action  of  various  wave  lengths  upon  the  color  nerves. 
—  According  to  the  Young-Helmholtz  theory  the  message  im- 
parted to  the  brain  by  each  set  of  nerves  is  entirely  indepen- 
dent of  the  nature  of  the  stimulus.  The  red  nerves,  however, 

*  Titchener,  An  Outline  of  Psychology,  p.  66. 
t  Rood,  Modern  Chromatics,  Chapter  IX. 


COLOR. 


ID/ 


are  most  strongly  affected  by  the  wave  lengths  at  the  red  end 
of  the  spectrum,  the  green  nerves  by  those  in  the  middle  of 
the  spectrum,  and  the  violet  nerves  by  the  shortest  visible 
waves. 

It  has  been  found  possible  to  isolate  these  primary  color  sen- 
sations and  to  determine  the  intensity  of  each  for  each  wave 
length  of  the  spectrum.  The  curves  in  Fig.  499  show  the 


HG  FE  D^CB 

t  £  S 

o  d  S 

>  m  o  >-  o: 

Fig.  499. 

result  of  such  measurements  made  by  Koenig.  It  will  be  seen 
that  all  the  rays  of  the  spectrum  are  capable  of  producing  in 
some  degree  the  sensation  of  red,  and  that  the  greater  portion 
of  them  have  some  effect  likewise  on  the  green  nerves.  Those 
which  affect  the  violet  nerves  appreciably  lie  between  the  mid- 
dle of  the  spectrum  and  the  violet  end. 

Contrast  effects.  —  When  two  colors  are  viewed  in  succession 
the  character  of  the  second  color  is  more  or  less  changed.  This 
action  is  called  a  contrast  effect.  The  set  of  nerves  most  affected 
by  the  first  color  seems  to  become  more  or  less  inactive  through 
fatigue,  which  results  in  the  preponderance  *  of  the  other  two 
primary  sensations  when  the  second  color  is  viewed.  The  fol- 
lowing example  will  make  this  clear. 

*  A  color  is  said  to  be  saturated  when  one  of  the  primary  sensations  greatly  pre- 
ponderates over  the  other  two.  The  mixture  of  white  light  with  a  colored  light 
reduces  the  saturation  by  bringing  all  three  sets  of  nerves  into  activity.  Any  process 
which  tends  to  isolate  a  single  primary  sensation  increases  the  saturation. 


I08  ELEMENTS   OF   PHYSICS. 

A  homogeneous  beam  of  light  of  a  wave  length,  for  example, 
corresponding  to  the  Fraunhofer  line  E,  will  stimulate  all  three 
sets  of  color  nerves.  The  resultant  sensation  will  be  composed 
of  a  powerful  primary  sensation  of  green,  a  weak  primary  sensa- 
tion of  red,  and  a  still  weaker  primary  sensation  of  violet.  The 
resultant  sensation  of  green  thus  produced  is  less  intense  than  it 
would  be  were  the  red  nerves  absent.  To  make  the  green  more 
vivid,  it  is  only  necessary  to  reduce  the  activity  of  the  red  nerves. 
This  may  be  done,  for  example,  by  wearing  spectacles  of  ruby 
glass  for  several  minutes ;  during  which  time  the  eyes  are 
exposed  to  bright  light.  Upon  removing  these  spectacles  the 
green  nerves,  which  have  been  protected  by  the  opacity  of  the 
ruby  glass  to  the  rays  which  most  strongly  affect  them,  will 
remain  active,  while  the  red  nerves  will  be  greatly  fatigued.  The 
result  is  a  greatly  increased  vivacity  and  intensity  of  all  sensa- 
tions of  green,  with  a  relative  loss  of  the  sensation  for  red. 
Extreme  fatigue  of  one  set  of  color  nerves  produces  tempo- 
rary partial  color  blindness.  Similar  effects  can  be  produced  by 
the  use  of  certain  drugs,  one  of  which,  santonine,  renders  the 
violet  nerve  temporarily  inactive.  The  excessive  use  of  tobacco 
sometimes  produces  partial  annihilation  of  the  activities  of  one 
or  more  of  these  color  nerves. 

729.  Peculiarities  of  dichroic  vision.  —  Color-blind  persons 
show  marked  peculiarities  in  their  classification  of  colors.  These 
peculiarities  are  taken  advantage  of  in  testing  for  color  blind- 
ness. They  are  described  in  Art.  730. 

The  appearance  of  the  spectrum  to  a  color-blind  person  is 
very  different  from  its  appearance  to  a  person  with  normal 
vision.  A  color-blind  person  has  two  primary  color  sensations. 
If  these  be  red  and  violet,  which  is  the  case  in  green  blindness, 
the  spectrum  is  made  up  of  these  two  sensations.  The  violet 
end  appears  very  much  as  it  would  to  the  normal  eye,  but  of  a 
higher  saturation.  The  red  end  is  also  more  intense  in  color, 
but  it  merges  into  white  instead  of  merging  into  yellow  and 


COLOR. 


109 


green.     The  center  of  the  spectrum  between  the  E  and  F  lines 
appears  to  such  individuals  of  a  neutral  shade. 

To  red-blind  individuals  the  spectrum  is  also  made  up  of  two 
tints ;  violet  at  the  violet  end  and  green  in  the  place  of 
red  at  the  red  end.  The  center  of  the  spectrum,  to  them,  is 
colorless.  The  red  end  of  the  spectrum,  which  appears  most 

G  F  E  D  C        B 


GREEN  BLIND 


WHITE 

Fig.  500. 


intense  to  green-blind  persons  and  to  persons  with  trichroic 
vision,  is  invisible  to  the  red  blind  ;  so  that  the  limit  of  visibility 
at  this  end  of  the  spectrum  is  shifted  towards  the  shorter  wave 
In  Fig.  500  the  arrangement  of  colors  in  the  spec- 


lengths. 


40 


20 


5000 


6000 
Fig.  501. 


7000 

RED 


trum  as  viewed    by  normal  and  by  color-blind    individuals    is 
indicated. 

One  method  of  studying  color  blindness  is  by  means  of  the 
curve  of   luminosity.     Figure   501    gives    such   a   curve,    from 


IIO  ELEMENTS   OF   PHYSICS. 

measurements  by  Ferry,*  together  with  the  curve  for  a  normal 
eye.     The  observer  was  partially  red  blind. 

It  will  be  seen,  from  the  figure,  that  the  luminosity  of  red 
and  yellow  light  was  below  the  normal,  and  that  in  the  green 
the  curve  joins  that  for  the  normal  eye.  In  the  blue  and  violet 
the  luminosity  was  normal. 

730.  Testing  for  color  blindness.  —  Since  all  our  systems  of 
signaling,  both  upon  railways  and  at  sea,  depend  upon  the 
recognition  of  colored  lights,  the  character  of  the  color  vision 
of  employees  is  a  matter  of  the  utmost  practical  importance. 
It  is  fortunately  possible  to  detect  dichroic  vision  with  cer- 
tainty even  where  the  existence  of  the  peculiarity  is  unsus- 
pected by  the  subject  himself.  The  simplest  satisfactory 
method  of  performing  such  tests  was  invented  by  Holmgren  of 
Upsala,  after  the  occurrence  of  a  terrible  railway  accident  due 
to  the  color  blindness  of  an  employee.  The  Holmgren  appara- 
tus consists  simply  of  a  collection  of  colored  worsteds,  which 
includes  a  variety  of  reds,  greens,  and  purples,  together  with 
tints  made  up  of  an  admixture  of  these  and  of  a  large  number 
of  neutral  grays  of  various  degrees  of  brightness.  There  are 
also  two  skeins,  called  the  confusion  samples.  One  of  these  is 
a  pale  apple  green :  its  color  corresponds  most  nearly  to  the 
central  portion  of  the  spectrum,  which  in  the  case  of  both  red- 
and  green-blind  persons  constitutes  the  neutral  region  described 
in  Art.  429. 

Both  red-  and  green-blind  subjects  inevitably  select,  as  closely 
corresponding  to  this  sample,  a  variety  of  neutral  worsteds. 
The  other  confusion  sample  is  a  light  magenta,  which  is  an 
admixture,  in  nearly  equal  parts,  of  red  and  violet,  with  much 
white.  Red-blind  persons  when  asked  to  pick  out  those  worsteds 
which  agree  most  nearly  in  color  with  this  sample,  invariably 
select  violets  and  blues.  The  selections  made  by  persons  of 
dichroic  vision,  when  attempting  to  follow  their  own  judgment 

*  American  Journal  of  Science,  Vol.  44  (1892). 


COLOR.  r  !  j 

in  matching  these  confusion  samples  in  the  Holmgren  test,  are 
very  striking  to  an  observer  with  normal  vision.  A  university 
student,  who  was  red  blind,  but  who  showed  great  power  of 
discrimination  in  his  choice  of  colors,  as  viewed  from  his  own 
standard,  and  who  was  fully  aware  of  the  nature  of  his  color 
sense,  made  the  following  selections  of  the  worsteds  which 
seemed  to  him  most  nearly  related  to  the  two  confusion  samples. 

Of  the  twelve  skeins  of  worsted  selected  to  go  with  the 
apple-green  sample,  only  one  contained  any  green  pigment. 
The  remainder  were  very  pale  yellows  and  browns,  almost  with- 
out coloring  matter,  and  two  skeins  of  very  pale  rose  color. 

All  the  worsteds  selected  on  account  of  their  resemblance  to 
the  magenta  sample  were  purples  in  which  blue  predominated 
or  nearly  pure  blues.  With  the  scarlet  were  placed  a  few  other 
nearly  pure  reds,  together  with  a  much  larger  number  of  dark 
browns  and  greens. 

Surprising  as  these  selections  seem  to  one  accustomed  to  the 
trichroic  color  system,  they  are  all  readily  accounted  for  under 
the  Young-Helmholtz  theory  of  colors. 


CHAPTER   X. 
PHOTOMETRY. 

731.  Definitions. — A  luminous  object  is  one  which  shines  in 
the  dark.     The  light  given  off   by  a  luminous  body  when  it 
strikes  surrounding  bodies  is  reflected  by  them  to  the  eye,  and 
they  become  visible.     These  bodies  are  said  to  be  illuminated. 
A  transparent  body  is  one  which,  like  glass,  permits  light  to 
pass  through  it.     A  body  which,  like  celluloid  or  horn,  appears 
milky  by  transmitted  light  is  said  to  be  translucent.     An  opaque 
body  is  one  which  does  not  permit  light  to  pass  through  it. 
Extremely  thin  layers  of  almost  all  opaque  substances,  such  as 
the  metals,  are  transparent. 

732.  The  law  of  inverse  squares;    brightness.  —  Imagine  a 
spherical  surface  of  radius  d  described  about  a  luminous  body 
at  its  center.     Light  from   the  luminous  body  is  distributed 
over  this  spherical  surface.     The  intensity  of  illumination  of 
the  surface,  or  of  any  part  of  it,  is  therefore  inversely  propor- 
tional to  its  area,  or  inversely  proportional  to  the  square  of  its 
radius.     We  have,  therefore, 

/=§.  (338) 

in  which  /  is  the  intensity  of  illumination  at  a  place  distant 
d  from  the  luminous  body,  and  B  is  the  proportionality  factor. 
This  quantity  B  is  called  the  brightness  of  the  luminous  body. 

733.  Standards  of  brightness,  or  light  standards.  —  In  all  the 

photometric  operations  to  be  described  in  this  chapter,  artificial 

112 


PHOTOMETRY. 


light  sources  —  such  as  gas  flames,  incandescent  lamps,  etc.  — 
are  compared  with  some  source  of  light  which  has  been  taken 
as  a  standard.  The  earliest  standard  of  brightness  was  the 
candle ;  and  this  standard,  although  it  has  been  shown  to  be 
one  of  the  least  constant  of  artificial  sources  of  light,  is  still 
recognized  as  the  official  standard  in  Great  Britain,  in  Germany, 
and  in  the  United  States. 

The  British  standard  candle  is  a  sperm  candle,  weighing  six 
to  the  pound,  and  burning  120  grains  per  hour.  Many  laborious 
investigations  of  the  behavior  of  this  standard  have  been  made, 
and  it  has  been  shown  that  it  fluctuates  through  a  very  great 
range  (more  than  20  %).  The  brightness  of  the  candle  depends 
upon  the  length  and  shape  of  the  wick,  the  height  of  the  flame, 
and  even  upon  the  temperature  and  humidity  of  the  air  of  the 
photometer  room. 

The  German  standard  candle  is  made  of  paraffin.  It  has  a 
uniform  diameter  of  two  centimeters.  The  wick  of  the  German 
candle  is  trimmed  until  the  height  of  the  flame  is  precisely  five 
centimeters,  when  measurements  are  taken.  Investigations  of 
this  standard  have  shown  that  even  with  these  precautions  it 
is  subject  to  very  considerable  fluctuation. 


I- 
E35 


30 


10m. 


20m. 


TIME 
Fig.  502.— The  British  candle. 


By  means  of  a  bolometer  (see  Chapter  XII.)  and  a  sensitive 
galvanometer,  it  is  possible  to  follow  the  continual  fluctuations 
of  a  candle  flame  from  moment  to  moment,  and  to  draw  a 
curve  showing  graphically  the  nature  of  these  variations.  Ex- 


ELEMENTS   OF   PHYSICS. 


tensive  measurements  of  this  kind  have  been  made  by  Sharp 
and  Turnbull.*  Figure  502  gives  a  portion  of  such  a  curve 
plotted  from  measurements  upon  a  British  standard 
candle.  The  interval  included  in  the  diagram  is 
half  an  hour.  It  will  be  seen  that  there  are 
numerous  and  very  sudden  diminutions  in  intensity. 
Various  other  flames  and  light  sources  have  been 
used  as  standards.  One  of  the  earliest  of  these 
was  the  Carcel  lamp,  which  was  invented  in  France. 
It  is  a  lamp  with  a  cylindrical  Argand  burner  and 
central  draught.  The  draught  is  maintained  by 
clockwork.  The  fuel  used  in  this  lamp  is  colza  oil. 
Another  standard  which  has  been  extensively 
used  is  the  Methven  screen.  In  this  apparatus  an 
ordinary  argand  gas  burner  is  provided  with  an 
opaque  screen  (Fig.  503),  through  which  a  rectan- 


Fig.  503. 


gular  opening  is  cut  of  such  size  as  to  reduce  the  flame  to 
two  candle  power.  The  opening  transmits  light  only  from 
the  central  and  brightest  portions  of  the  gas  flame.  The 
variations  of  such  a  standard  are  those  due  to  the  varying 
qualities  of  the  illuminating  gas,  and  to  the  varying  pressures 
under  which  it  is  consumed. 


10  m. 


30m. 


TIME 
Fig.  504.  —The  Methven  standard. 

Figure  504  shows  the  curve  obtained  with  this  lamp.  It  will 
be  seen  that  the  flame  was  subject  to  many  large  and  rapid 
variations. 

*  Physical  Review,  Vol.  II.,  p.  I. 


V^ftML^ 


PHOTOMETRY. 


In  Germany  the  candle  has  been  largely  supplanted  as  a 
standard  by  the  Hefner-Alteneck  lamp.  This  is  a  simple 
metal  lamp,  .burning  amyl  acetate.  It  is  arranged  so  that  the 
height  of  the  flame  can  be  accurately  adjusted  and  measured. 
It  has  been  found  that  this  standard  is  reliable  to  within  two 
per  cent,  which  is  a  better  result  than  has  been  obtained,  thus 
far,  with  other  flames.  Figure  505  shows  a  typical  bolometric 


45 


35 


10  m. 


20  m. 

TIME 


30  m. 


Fig.  505.  —  The  Hefner  lamp. 


curve  for  the  Hefner  lamp.  During  the  half  hour  included  in 
this  test,  the  brightness  of  the  flame  was  very  nearly  constant. 

The  Violle  standard.  —  Violle  has  proposed,  as  a  standard  of 
brightness,  the  light  emitted  by  a  square  centimeter  of  surface 
of  platinum  at  its  melting  point.  It  has  been  found  imprac- 
ticable to  put  this  standard  into  general  use. 

The  glow  lamp.  —  The  glow  lamp,  when  supplied  with  con- 
stant current,  is  free  from  the  difficulties  which  interfere  with 
the  accuracy  of  flame  standards.  It  has  been  widely  adopted 
in  photometric  work  as  a  comparison  standard,  but  thus  far  it 
has  been  found  impossible  to  produce  glow  lamps  which  do  not 
change  in  brightness  with  age.  It  has  not  been  found  practi- 
cable to  so  specify  the  character  of  the  parts  of  a  glow  lamp 
that  one  could  be  manufactured  which,  when  provided  with  a 
given  current,  would  give  the  specified  candle  power. 

734.  Intensity  of  illumination;  intrinsic  brightness.  —  When 
B  (equation  338)  is  one  candle  and  d  is  one  meter,  then  /  is 

I   (  -  -  ).     This  intensity  of  illumination,  namely  the  illumi- 
Vmeter2/ 


H6  ELEMENTS   OF   PHYSICS. 

nation  by  a  standard  candle  at  a  distance  of  one  meter,  is  occa- 
sionally used  as  a  unit  for  expressing  intensity  of  illumination. 
Thus  the  least  intensity  of  illumination  required  for  comforta- 

candles 
(meter)' 


ble   reading  is  about  4   -         — -   (read  four  candles-per-meter- 


square^) 

The  intrinsic  brightness  of  a  luminous  body  is  defined  as  its 
total  brightness,  B,  divided  by  the  area  of  its  shining  sur- 
face. Thus  the  intrinsic  brightness  of  a  candle  flame  is  about 

o.i  es  ;  of  an  Argand  gas  flame  it  is  about  one  candle  per 

cm. 

square  centimeter;  of  a  glow  lamp  it  is  from   10  to  20  can    2es> 

and  for  the  crater  of  the  arc  lamp  it  is  as  much  as  20,000  ^^  —  ^- 

cm.2 

735.  Bouguer's  principle.  —  It  was  first  shown  by  Bougtier 
(1726)  that  the  relative  brightness  of  two  lamps  might  be 
determined  by  measuring  the  distances  at  which  those  lamps 
would  give  equal  intensities  of  illumination  upon  a  screen. 
This  principle  is  the  basis  of  most  practical  photometric 
methods.  Consider  two  lamps.  Let  B  be  the  brightness  of 
one,  and  d  the  distance  from  a  screen  at  which  it  gives  an  illu- 
mination of  intensity  /.  Then  from  equation  338  we  have 


Let  B1  be  the  brightness  of  the  other  lamp,  and  d'  the  dis- 
tance from  a  screen  at  which  it  gives  the  same  intensity  of  illu- 
mination ;  then 


From  these  equations  we  have,  upon  eliminating  /, 

B'  =  ^B.  (339) 

736.    Limitations  of   simple   photometry.  —  The   photometric 
comparison  of  lamps  which  give  light  of  identically  the  same 


PHOTOMETRY.  II7 

composition  is  called  simple  photometry.  Lights  which  differ 
but  slightly  in  composition,  or  tint,  cannot  be  compared  with 
any  accuracy  by  the  methods  of  simple  photometry,  and  when 
the  difference  in  tint  is  distinctly  appreciable,  the  comparison 
cannot  be  made  at  all.  The  relative  brightness  of  two  lamps  of 
different  tint  may  be  estimated  with  some  accuracy  by  a  method 
devised  by  Crova,  and  by  a  device  called  the  flicker  photometer. 
The  complete  comparison  of  lights  of  different  composition  can 
be  made  only  with  the  help  of  the  spectrophotometer. 

737.  The  shadow  photometer  was  devised  by  Bouguer,  and  was 
used  later  in  a  modified  form  by  Lambert,  who  wrote  an  extended 
treatise  on  photometry  *  in  1760.     It  was  also  used  by  Rumford, 
and  is  often  called  Rumford 's  Photometer.    The  shadow  photom- 
eter is  an  arrangement  in  which  the  two  sources   of  light  to 
be  compared  cast  overlapping  shadows  upon  a  screen,  as  shown 
by  the  overlapping  squares,  A  and  By  in  Fig.  506.     The  portion 
C  of  the  screen  which  is  common  to  both 

shadows  is  illuminated  by  neither  source, 
while  the  remaining  portions  of  A  and  B 
are  illuminated  each  by  a  single  source. 
The  lamps  are  moved  to  such  distances 
from  the  screen  as  to  bring  the  contiguous 

portions  of  A  and  B,  namely  the  portions  d  and  e,  to  equal  bright- 
ness. The  distances  of  the  lamps  from  the  screen  are  measured, 
and  the  brightness  of  one  lamp  in  terms  of  the  other  is  then 
given  by  equation  339. 

738.  The  Bunsen  photometer. — This  instrument,  which  has 
been  more  widely  used  in  practical  photometry  than  any  other, 
depends  upon  Bouguer's  principle  and  is  peculiar  in  the  method 
employed  in  judging  the  equality  of  illumination  produced  upon 
the  screen  by  the  two  lamps  to  be  compared.     The  screen  is 

*  Photometria  sive  de  mensura  et  gradibus  luminis,  colorum  et  umbrae.     Aug. 
Vind.  1760. 


U8  ELEMENTS   OF   PHYSICS. 

made  of  unsized  paper  thick  enough  to  be  opaque.  All  but 
a  central  spot  is  made  translucent  by  soaking  with  oil  or  paraf- 
fin. If  this  screen  be  observed  by  reflected  light,  the  unoiled 
portion  will  appear  bright  and  the  oiled  portion  will  appear 
dark.  A  considerable  proportion  of  the  light  striking  the  oiled 
surface  penetrates  the  same  and  is  transmitted,  and  when  the 
screen  is  viewed  by  transmitted  light  it  presents  the  precisely 
opposite  appearance ;  the  central  spot  is  dark  upon  a  bright 
background.  The  appearance  of  the  Bunsen  screen  under 
these  conditions  is  illustrated  in  Fig.  507.  When  the  two 

sides  of  such  a  screen  are  equally  illu- 
minated, the  central  spot  appears  of 
the  same  brightness  as  the  surround- 
ing oiled  surface.  The  effect  of  the 
oil  is  to  vary  slightly  the  reflecting 
power  of  the  surface  ;  so  that  complete 

identity  of  appearance  is  never  secured :  the  spot  of  light  is, 
therefore,  not  absolutely  lost  to  view  even  when  the  illumina- 
tion from  the  two  sides  is  the  same.  Under  such  conditions, 
however,  the  two  faces  of  the  paper  are  identical  in  appear- 
ance. In  practice  the  screen  of  the  Bunsen  photometer  is 
placed  between  two  oblique  mirrors,  within  a  box  or  carriage 
which  slides  or  rolls  upon  a  track,  at  the  ends  of  which  are 
situated  the  light  sources  to  be  compared.  The  track  is 
called  the  photometer  bar.  It  is  provided  with  a  scale,  usually 
of  1000  equal  parts.  The  carriage  is  moved  along  the  bar 
until  equal  illumination  upon  the  two  sides  of  the  screen  has 
been  secured,  when  the  distances  d^  and  d^  between  the  disk 
and  the  sources  of  light  are  read  upon  the  scale.  Equation  339 
then  gives  the  brightness  of  one  lamp  in  terms  of  the  other. 

The  object  of  the  two  mirrors  between  which  the  screen  is 
mounted,  as  shown  in  Fig.  508,  is  to  enable  the  observer  to  see 
the  images  of  both  sides  of  the  screen  simultaneously.  The 
observation  consists  in  so  placing  the  carriage  that  the  two 
images  shall  be  identical  in  appearance. 


PHOTOMETRY. 


119 


The  accurate  use  of  the  Bunsen  photometer  supposes  identity 
in  the  color  of  the  two  light  sources  to  be  compared.  As  in 
the  case  of  the  shadow  photometer,  and  of  all 
instruments  based  upon  the  principle  of  the 
equality  of  two  fields  of  view,  differences  in 
the  color  of  the  light  reflected  from  the  two 
surfaces  of  the  screen,  even  when  very  slight, 
vitiate  the  judgment  as  to  their  relative 
brightness. 


Fig.  508. 


739.  The  Lummer-Brodhun  photometer.  —  This  instrument  is 
a  modification  of  the  Bunsen  photometer,  which  has  been  found 
to  possess  one  great  advantage.  In  the  attempt  to  observe  the 
two  images  of  the  Bunsen  disk  simultaneously  all  experimenters 
with  that  instrument  acquire  the  habit  of  observing  with  the 
two  eyes  independently.  They  view  the  right-hand  image  with 
the  right  eye,  and  the  left-hand  image  with  the  left  eye.  Owing 
to  differences  in  the  sensitiveness  of  the  two  eyes,  the  images 
appear  to  be  equally  bright  when 
they  are  not  so  in  reality.  The 
result  is  a  false  setting  of  the  pho-  s, 
tometer,  which  is  persistent  and 
of  nearly  constant  value  with  each 
individual.  It  is  found  that  a  very 
large  proportion  of  observers  set 
the  disk  to  the  left  of  its  true 
position  ;  as  though  the  right  eye 
were  the  more  sensitive  organ. 
A  few  observers  have  the  opposite 
tendency.  Observations  are  made 
with  one  eye  in  the  Lummer- 
Brodhun  photometer,  and  this  personal  error  is  avoided. 

In  this  instrument  light  from  the  two  sources  S1  S2,  Fig.  509. 
falls  upon  the  two  faces  of  an  opaque  screen  AB,  which  is 
whitened  with  magnesium  oxide. 


120 


ELEMENTS   OF   PHYSICS. 


FROM  S2 


FROM  S2 


LIGHT  FROM  S, 


LIGHT  FROM  82 


The  observer  at  the  small  telescope  sees  portions  of  the  two 

sides  of  this  opaque  screen  side  by  side  in  the  same  field  of  view. 

The  cube  of  glass  CD  is  made  of  two  right-angled  prisms, 

the  diagonal  face  of  one  of  which  is  partly  cut  away,  as  shown 

in  Fig.  510.  These  prisms  are 
then  cemented  together  with 
Canada  balsam  so  that  light  from 
the  mirror  M2  passes  through  the 
cemented  portion  unobstructed, 
while  it  is  totally  reflected  else- 
where. The  light  from  the  mirror 
FROM  8l  j^  is  totaiiy  reflected  from  the 

"Sz  free  portions  of  the  diagonal  plane 

Fig-  510<  and  passes  through  the  cemented 

portion    unobstructed.       The    cemented   portion    is    commonly 
circular,  and  the  appearance  of  the  field  of  view  of   the  tele- 
scope  is    shown    in    Fig.   511.      The 
whole   arrangement    is    moved   along 
the   photometer   bar,  as   in  the  case 
of    Bunsen's    photometer,    until    the 
field  of  view  of  the  telescope  is  uni- 
formly illuminated.     The  distances  of 
Fig-511>  the  screen  AB  from  the  sources   Sl 

and  S%  are  then  observed,  and  the  brightness  of  one  source 
in  terms  of  the  brightness  of  the  other  is  given  by  equation 
339.  The  Lummer-Brodhun  arrangement  is  slightly  more  sensi- 
tive than  that  of  Bunsen. 

740.  Distribution  of  brightness.  —  Light  from  a  single  point 
travels  outward  in  spherical  waves,  and  the  intensity  of  illumi- 
nation at  a  given  distance  from  the  point  is  in  all  directions  the 
same.  In  the  case  of  the  light  from  the  various  flames  used  in 
artificial  illumination,  from  the  filaments  of  glow  lamps,  and  from 
the  carbons  of  the  arc  light,  the  distribution  is  not  uniform. 
The  study  of  the  distribution  of  brightness  is  made  by  turn- 


PHOTOMETRY. 


121 


ing  the  flame  about  into  various  positions  and  measuring  its 
brightness.  The  distribution  of  light  in  a  given  plane  is  then 
represented  by  means  of  a  polar  diagram  in  which  the  radius 
vector  gives  the  brightness  at  all  angles.  If  the  flat  flame  of 
a  bat's-wing  gas  burner  be  thus  measured  in  a  horizontal  plane, 
it  will  be  found  that  the  distribution  is  uniform,  and  that  the 
curve  is  a  circle.  The  fact  that  such  a  flame  is  as  bright  when 
viewed  edgewise  as  when  its  entire  breadth  is  exposed,  shows 
that  the  light  comes  from  a  comparatively  small  number  of 
isolated  particles  of  glowing  carbon,  which  are  so  few  and  far 
between  that  they  do  not,  to  any  considerable  extent,  screen 
each  other.  If  the  flame  from  a  richer 
fuel  such  as  petroleum  be  tested  in  like 
manner,  it  will  be  found  that  the  flame  is 
not  so  fully  transparent.  Figure  512  shows 
the  curve  of  horizontal  distribution  in  the 
case  of  the  flame  of  an  ordinary  petroleum 
lamp.  The  falling  off  of  brightness  in 
the  plane  of  the  flame  shows  that  there 
is  a  considerable  screening  action.  The 
length  of  the  radius  vector  of  the  curve  in  any  direction  repre- 
sents the  candle  power  of  the  lamp  in  that  direction. 

The  distribution  of  bright- 
ness in  the  case  of  the  arc 
lamp  is  very  far  from  uniform. 
In  commercial  direct-current 
arc  lamps  the  current  is  always 
sent  through  the  lamp  in  such 
a  direction  that  the  crater  is 
formed  at  the  end  of  the  upper 
carbon.  The  vertical  distribu- 

£   ,     .     ,   .  .       ..  .  Fig.  513.  Fig.  514. 

tion  of  brightness  in  this  case 

is  shown  by  the  curve  in  Fig.  513.  The  vertical  distribution 
of  brightness  in  case  of  the  alternating  current  arc  is  shown 
by  the  curve  in  Fig.  514.  The  carbons  in  an  alternating  cur- 
rent arc  are  equally  heated. 


Fig.  512. 


122  ELEMENTS   OF   PHYSICS. 

The  distribution  of  brightness  of  an  arc  lamp  varies  in  a 
rapid  and  irregular  manner  on  account  of  the  shifting  of  the 
arc.  Figures  513  and  514  show  the  average  distribution  of 
light ;  the  momentary  distribution  may  be  very  different. 

741.    The   photometry   of   lights   differing   in   composition.  — 

The  rigorous  method  for  comparing  lights  of  different  compo- 
sitions is  by  means  of  the  spectrophotometer,  which  has  been 
described  in  Art.  703.  In  addition  to  the  method,  there 
explained,  for  determining  the  brightness  of  each  part  of  the 
spectrum  of  the  light  which  is  being  studied,  in  terms  of  the 
brightness  of  the  same  part  of  the  spectrum  of  a  standard  light, 
the  following  methods  are  frequently  employed. 

(a)  Vierordt's  slit.  — This  is  the  earliest  device  used  in  spectro- 
photometry.     The  two  ends  of  the  slit  of  the  spectrometer  are 
arranged  to  be  opened  or  closed  independently  of  each  other  by 
means  of  micrometer  screws.     The  instrument  is  so  arranged 
that  the  beams  of  light  from  the  sources  to  be  compared  pass 
through  these  halves  of  the  slit.     The  spectra  are  brought  to 
equality  at  a  given    region    by  opening  that    half  of  the  slit 
through  which  the  weaker  beam  enters.     The  relative  bright- 
ness of  the  sources  for  the  given  region  of  the  spectrum  is 
known  from  the  relative  widths  of  the  slit  halves.     This  is  an 
exceedingly  simple  and  convenient  method,  but  its  usefulness 
is  limited  to  cases  where  the  ratio  of  intensities  is  not  large. 
Such  a  slit  cannot  be  made  very  wide  without  affecting  the 
purity  of  the   colors  of   the   spectrum   by  overlapping  of  the 
images.     As  soon  as  this  change  in  the  quality  of  the  spec- 
trum begins  to  be  noticeable,  the  limit  of  usefulness  of  the 
apparatus  has  been  reached. 

(b)  The  method  of  the  Nicol  prisms.  —  In  this,  which  is  the 
method  most  frequently  employed,  a  pair  of  Nicol  prisms  are 
mounted,  as  in  the  polariscope  (Art.  751),  and  are  placed  in 
the  path  of  the  brighter  of  the  two  beams  of  light.     This  beam 
enters  one  end  of  the  slit  of  the  spectrophotometer.     One  of 


PHOTOMETRY.  I23 

the  Nicol  prisms  is  turned  until  the  spectra  are  of  equal  inten- 
sity at  a  given  region.  The  angle  between  the  principal  planes 
of  the  prisms  is  then  observed,  and  the  relative  brightness  of 
the  sources  for  the  given  spectral  region  is  calculated  as  ex- 
plained in  Art.  750. 

742.  Approximate  methods.  —  The  following  approximate 
methods  do  not  give  exact  values.  They  consist  in  the  employ- 
ment of  some  simple  device  for  overcoming  more  or  less 
completely  the  difficulties  arising  from  the  difference  in  the 
color  of  the  lights  to  be  compared. 

(a)  Crovas  method.  —  When  incandescent  carbon,  which  is 
the  glowing  material  in  nearly  all  artificial  illuminants,  rises 
in  temperature,  all  the  wave  lengths  of  the  spectrum  increase 
in  brightness  simultaneously.  The  rate  of  increase  is  small- 
est in  the  red,  and  becomes  continually  greater  as  the  wave 
length  diminishes.  The  result  is  a  gradual  change  in  the 
composition  of  the  light  which  the  glowing  carbon  emits. 
Since  the  rate  of  increase  of  intensity  varies  continuously 
from  red  to  violet,  there  must,  in  every  case,  be  some  par- 
ticular wave  length  the  change  of  which  is  the  same  as  the 
change  in  brightness  of  the  light  taken  as  a  whole.  Having 
determined  the  portion  of  the  spectrum  for  which  this  is  true, 
we  may  confine  our  attention  entirely  to  that.  From  the  rela- 
tive brightness  of  that  particular  region  we  thus  obtain  the 
relative  brightness  of  the  sources  to  be  compared. 

This  method  was  suggested  by  Crova,  who  found  the  proper 
wave  length  for  such  measurements  to  be  in  the  yellow  (5800 
Angstrom  units).  Crova's  method  consists  in  the  use  of  a 
spectrophotometer,  by  means  of  which  the  spectra  of  the  two 
sources  are  brought  to  equal  brightness  in  the  yellow.  It  is 
applicable  only  in  cases  where  the  law  of  radiation  of  the 
material  in  the  two  sources  of  light  is  the  same,  and  where 
the  difference  in  temperature  is  not  very  great.  In  more 
extreme  cases  it  affords  only  a  rough  approximation  to  the 
true  photometric  ratio. 


124 


ELEMENTS   OF   PHYSICS. 


(b)  The  flicker  photometer  is  an  arrangement  by  means  of 
which  the  two  sides  of  a  photometer  screen  are  brought  into 
the  same  field  of  view  in  rapid  succession.  If  the  frequency 
of  interchange  is  very  rapid,  the  illumination  will  appear  uni- 
form whatever  the  relative  brightness  of  the  two  sides  of  the 
screen  may  be;  but  if  the  frequency  is  only  moderately  rapid, 
the  flickering  will  cease  only  when  the  light  shining  upon  the 
two  faces  of  the  screen  brings  them  to  the  same  luminosity. 
This  fact  was  discovered  by  Rood.  The  device  employed  by 
Whitman,*  who  has  based  a  successful  photometric  method 
upon  it,  is  as  follows  : 


T      T 


Fig.  515. 


Fig.  516. 


A  white  cardboard  disk,  A,  Figs.  515  and  516,  is  mounted 
upon  the  axis  //,  and  a  stationary  piece  of  the  same  cardboard 
is  placed  at  B.  When  the  disk  A  is  set  in  rapid  rotation,  the 
cardboard  B  and  the  wing  of  the  disk  are  seen  in  rapid  suc- 
cession by  the  eye  placed  at  e.  The  wing  of  the  disk  is  il- 
luminated by  the  source  Sv  and  the  stationary  cardboard  is 
illuminated  by  the  source  52.  The  distances  of  the  sources 
are  adjusted  until  the  "sense  of  flickering"  disappears. 

*  F.  P.  Whitman,  Physical  Review,  Vol.  III.,  p.  241. 


CHAPTER   XI. 
POLARIZATION   AND   DOUBLE    REFRACTION. 

743.  The  side  aspect  of  transverse  waves.  —  Consider  a 
stretched  cord,  for  example  a  violin  string.  Such  a  string  set 
vibrating  longitudinally  by  rubbing  it  lengthwise  presents  iden- 
tically the  same  appearance  from  whatever  side  it  is  viewed. 
If  the  string  passes  loosely  through  a  slit  in  a  card,  the  longi- 
tudinal waves  (vibrations)  are  affected,  if  at  all,  in  a  similar  man- 
ner whatever  the  direction  of  the  slit.  The  transverse  waves,  on 
the  other  hand,  such  as  are  produced  by  the  action  of  a  violin 
bow  drawn  across  the  string,  pass  the  card  freely  if  the  slit  is 
parallel  to  the  direction  in  which  the  particles  vibrate,  but  they 
are  stopped  by  the  card  if  the  slit  is  perpendicular  to  the  vibra- 
tions. 

The  transverse  vibrations  of  the  string  may  be  such  that 
each  point  of  the  string  describes  a  circle,  the  plane  of  which 
is  perpendicular  to  the  string,  or  they  may  change  rapidly  and 
irregularly  from  one  direction  to  another.  In  either  case  the 
action  of  the  slit,  whatever  its  direction,  is  the  same,  so  far  as 
the  intensity  of  the  transmitted  waves  is  concerned.  Trans- 
verse waves,  whatever  the  character  of  the  vibration,  which 
pass  through  a  slit  in  one  card,  would  be  entirely  stopped  by 
a  second  card,  the  slit  in  which  is  held  at  right  angles  to  the 
slit  in  the  first. 

Transverse  waves  in  which  the  vibration  takes  place  continu- 
ously in  the  same  direction,  that  is  in  the  same  plane,  are  said 
to  \$£,  plane  polarized  or  svcw^ky  polarized.  When  the  vibrations 
are  irregular,  but  those  in  a  certain  direction  predominate,  the 

I25 


126  ELEMENTS   OF   PHYSICS. 

waves  are  said  to  \>e  partially  polarized.  When  the  particles  of 
the  medium  describe  circular  or  elliptical  paths,  the  waves  are 
said  to  be  respectively  circularly  or  elliptically  polarized. 

744.  The  optical    behavior   of  tourmaline  crystals ;  polarized 
light.  —  A  plate  of  this  mineral  cut  parallel  to  the  axis  of  the 
crystal  transmits  only  a  portion  of  the  light  which  falls  upon 
it.     The  light  which  passes  through  such  a  plate  passes  freely 
through  a  similar  plate  when  the  axes  of  the  plates  are  parallel, 
and  is  shut  off  entirely  when   the   axes  of  the  plates  are  at 
right  angles.     The  beam  of  light  transmitted  by  the  first  plate 
is  polarized,  as  is  evident  from  the  side   properties  which  it 
exhibits  with  respect  to  the  second  plate.     When  the  second 

plate  is  slowly  turned  about  an  axis 
parallel  to  the  beam  of  light,  the 
intensity  of  the  transmitted  beam 
changes  slowly  from  maximum  in- 
tensity when  the  axes  of  the  plates 

are  parallel  to  zero  intensity  when  the  axes  are  crossed.  This 
is  shown  by  the  shading  in  Fig.  517.  The  fact  that  light  can 
be  polarized  shows  that  light  waves  are  transverse  waves. 

745.  Polarization  of  light  by  reflection.  —  Light  which  is  re- 
flected from  a  polished  non-metallic  surface  is  polarized.     In 
general  such  a  reflected  beam  of  light  is  only  partially  polar- 

ized  (i.e.  it  cannot  be  completely  shut 
off  by  a  plate  of  tourmaline).  The  de- 
gree of  polarization  varies  with  the  angle 
of  incidence.  At  normal  incidence  the 
reflected  beam  is  not  at  all  polarized. 
As  the  incidence  becomes  more  oblique, 
the  degree  of  polarization  increases, 

POLARIZE  BEAM  reaches  a  maximum  when  the  reflected 

and  refracted  beams  are  at  right  angles 
Flg<  518-  (Brewster),  and  then  decreases.  The 

angle  of  incidence  for  which  the  degree  of  polarization  of  the 


POLARIZATION   AND   DOUBLE   REFRACTION. 


127 


reflected  beam  is  a  maximum  is  called  the  polarizing  angle. 
Its  tangent  is  equal  to  the  refractive  index  of  the  reflecting 
substance.  For  glass  the  polarizing  angle  is  about  57°  ;  for 
pure  water  it  is  53°  11'.  Substances  of  which  the  refractive 
index  is  about  1.46  give  complete  polarization  by  reflection 
at  the  polarizing  angle.  Figure  518  shows  a  glass  plate  ar- 
ranged to  give  a  beam  of  completely  polarized  light  by  reflec- 
tion. 

746.  Plane  of  polarization. — That  plane  passing  through  the 
reflected  beam  which  is  perpendicular  to  the  reflecting  surface 
is  called  the  plane  of  polarization  of  the  reflected  beam.     The 
vibrations  of  plane  polarized  light  must  be  either  parallel  to  or 
perpendicular  to  this  plane  of  polarization  thus  conventionally 
defined.     According  to  the  theory  of  Fresnel,  the  vibrations  of 
the  medium  are  perpendicular  to  this  plane.     According  to  the 
electro-magnetic  theory  of  light,  the  electric  force  is  perpendicu- 
lar to  and  the  magnetic  force  is  parallel  to  the  plane  of  polariza- 
tion.    The  character  of  plane  polarized  electro-magnetic  waves 
is  described  in  Art.  603  (Vol.  II.). 

747.  Reflection  of  polarized  light  from  a  polished  surface. — 

Consider  a  beam  of   polarized  light  incident  at  the  polarizing 


Fig.  519. 

angle  upon  a  glass  plate.     If  the  glass  plate  is  turned  about  the 
incident    beam   as   an   axis,    keeping    the   angle   of    incidence 


128 


ELEMENTS   OF   PHYSICS. 


il 


constant,  the  amount  of  light  which  is  reflected  varies  from 
a  maximum,  when  the  plane  of  polarization  of  the  incident 
beam  is  perpendicular  to  the  surface  of  the  glass,  to  zero  when 
the  glass  plate  is  turned  one  quarter  of  a  revolution  from  this 
position.  Figure  519  shows  the  two  positions  of  a  pair  of 
glass  plates,  A  and  B,  for  which  the  polarized  beam  reflected 
from  the  one  is  most  completely  reflected  by  the  other.  If 
either  plate  is  turned  one  quarter  of  a  revolution  about  the  ver- 
tical line  AB,  then  no  portion  of  the  polarized  beam  is  reflected 
from  the  plate  B. 

748.   Double  refraction.  —  Many  crystalline   substances  have 
the  property  of  dividing  a  beam  of  homogeneous  light  into  two 

beams  by  refraction.    This  phenomenon 
is    called   double   refraction.       All    allo- 
tropic  substances  are  doubly  refracting. 
The    crystalline   mineral,    Iceland    spar 
(calcium  carbonate),  separates   the  two 
refracted   beams  widely,  and   therefore 
shows  the  effect  very  distinctly. 
Consider  first  a  plate  of  glass,  AB  (Fig.  520).     A  beam  of  light 
from  e,  falling  upon  the  glass,  reaches  the  point  /,  and  if  /  is  a 

luminous  point,  it  will  be  seen  by 
an  eye  held  at  e,  as  though  it  were 
at  q.  If  the  plate  is  turned  about 

the  line  /  as   an   axis,  while   the 

point   /    is    stationary,    then   the 
/'  point  q  will  remain  stationary. 

A  beam  of  light  R  (Fig.  521), 
falling  upon  a  plate  AB  of  Ice- 
land spar,  is  broken  up  into  two 
rays,  and  a  point  /  sends  out  two  rays  o  and  x,  parallel 
to  O  and  X,  respectively,  which  enter  an  eye  at  e,  so  that 
the  point  /  is  seen  as  two  points  q  and  g'.  If  the  plate 
AB  is  rotated  about  the  line  /  as  an  axis,  one  of  the  images 


Fig.  520. 


POLARIZATION   AND   DOUBLE   REFRACTION. 


129 


q  remains  stationary,  just  as  if  AB  were  a  plate  of  glass,  and 
the  other  q'  moves  round  it.  The  ray  o  (or  O)  in  the  crystal 
corresponding  to  the  stationary  image  q  is  called  the  ordinary 
ray,  inasmuch  as  it  is  refracted  in  the  ordinary  way  (as  in  glass) ; 
and  the  ray  x  (or  X)  in  the  crystal  corresponding  to  the  moving 
image  q'  is  called  the  extraordinary  ray,  inasmuch  as  it  is  not 
refracted  in  the  ordinary  way.  Some  crystals  (biaxial  crystals) 
divide  a  beam  of  common  light  into  two  beams,  neither  of  which 
follows  the  ordinary  law  of  refraction. 

The  rays  r  and  r'  (Fig.  521)  are  completely  polarized,  and 
their  planes  of  polarization  are  at  right  angles.  This  may  be 
shown  by  holding  a  tourmaline  plate  (or  Nicol  prism)  before  the 
eye  at  e.  As  the  tourmaline  plate  is  turned,  one  and  then  the 
other  of  the  images  q  and  q'  becomes  invisible. 

749.  Huygens'  theory  of  double  refraction.  —  The  phenomena 
of  double  refraction  in  Iceland  spar  were  fully  analyzed  by 
Huygens.  He  assumed  two  secondary  wavelets  to  pass  out 
from  each  point  of  the  surface  of  a 
plate  of  Iceland  spar  as  an  incident 
wave  reached  that  point ;  one  of  these 
wavelets  being  a  sphere  and  the  other 

an   ellipsoid   of   revolution.     The    en-    C  [       (        P  ]       \  D 

velope  of  the  spherical  wavelets  deter- 
mines the  ordinary  refracted  wave,  as 
explained  in  Art.  635,  and  the  envelope 
of  the  ellipsoidal  wavelets  determines 

Fig.  522. 

the  extraordinary  refracted  wave. 

Let/  (Fig.  522)  be  a  center  of  disturbance  in  Iceland  spar; 
for  example,  a  point  of  a  wave  from  which  secondary  wavelets 
pass  out.  One  wavelet  is  a  sphere.  It  travels  out  from  p  at  a 

velocity   — - — -  as  great  as  the  velocity  of  light  in  air.     The 
1.658 

other  wavelet  is  an  oblate  spheroid  which  touches  the  sphere 

at  A  and  B,  and  of  which  the  major  diameter  CD  is       ^     times 

1.486 


130 


ELEMENTS   OF   PHYSICS. 


as  great  as  the  diameter  of  the  sphere.  The  axis  AB  of  the 
spheroid  is  parallel  to  the  axis  of  symmetry  of  the  crystal. 

The  axis  of  symmetry  of  a  crystal  of  Iceland  spar  is  called 
its  optic  axis. 

Any  plane  which  includes  the  optic  axis  is  called  a  principal 
plane. 

The  vibrations  (electric  force)  in  the  spheroidal  wavelet  are 
everywhere  in  the  principal  planes.  The  vibrations  in  the 
spherical  wavelet  are  everywhere  perpendicular  to  the  principal 
planes. 

Figure  523  shows  Huygens'  construction  for  the  ordinary 
and  extraordinary  waves  in  Iceland  spar.  Let  ww  be  the 
position  which  an  incident  wave  would  reach  at  a  given  instant 

in  a  homogeneous  medium.  Con- 
sider the  wavelets  which  ema- 
nated from  the  point  /  as  the 
wave  passed  that  point.  Let 
the  distance  from  /  to  ww  be 
R,  then  the  radius  of  the  spher- 

r> 

Fig.  523.  ical   wavelet    from  /    is  -— , 

r> 

and  the  major  semi-diameter  of  the  spheroidal  wavelet  is 

1.486 

The  axis  of  the  spheroidal  wavelet,  that  is,  the  optic  axis  of  the 
crystal,  is  supposed  to  be  given. 

The  line  o  represents  the  ordinary  wave,  and  r  the  ordinary 
ray.  The  line  x  represents  the  extraordinary  wave,  and  r'  the 
extraordinary  ray.  It  is  to  be  noticed  that  /  is  not  perpendicu- 
lar to  x.  In  fact,  the  extraordinary  wave  does  not  progress  in  a 
direction  at  "right  angles  to  its  front. 

When  the  ordinary  and  extraordinary  rays  coincide  with  the 
optic  axis,  there  is  no  double  refraction.  Crystals  like  Iceland 
spar,  which  have  only  one  direction  in  which  there  is  no  double 
refraction,  are  said  to  be  uniaxial  crystals.  Many  crystals  have 
two  such  directions  or  two  optic  axes.  Such  crystals  are  said 
to  be  biaxial. 


POLARIZATION   AND   DOUBLE   REFRACTION. 


750.  The  Nicol  prism.  —  A  beam  of  completely  polarized 
light  may  be  easily  obtained  by  reflection  from  a  glass  plate, 
as  explained  in  Art.  745.  A  much  more  convenient  arrange- 
ment, however,  for  obtaining  a  beam  of  completely  polarized 

light    is   the   Nicol  prism,    which A 

constructed     as     follows  :     A    /      \__        -\ -^— ^ 


is 


crystal  of  Iceland  spar  is  reduced 
to   the   form   shown  in  Fig.   524; 


Fig.  525. 


B 

Fig.  524. 

the  faces  of  this  rhomb  being  cleavage  planes  of  the  crys- 
tal. This  rhomb  is  divided  along  AB  perpendicular  to  the  plane 
of  the  paper.*  The  faces  along  AB  are  polished,  and  the  pieces 

cemented  together  with  a  thin 
layer  of  Canada  balsam,  the  re- 
fractive index  of  which  is  be- 
tween 1.658  and  1.486  (the  ordi- 
nary and  extraordinary  indices 
of  Iceland  spar).  An  incident 
ray  r  (Fig.  525)  is  broken  into  two  rays  by  the  spar,  and  the 
ordinary  ray  o  is  totally  reflected  from  the  layer  of  balsam  as 
shown.  The  extraordinary  ray  passes  on  through  and  is  com- 
pletely polarized. 

751.  The  action  of  a  Nicol  pnsm  on  a  beam  of 
polarized  light.  — Let  ABCD  (Fig.  526)  be  the  A- 
front  face  of  a  Nicol  prism,  and  let  the  line  a 
represent  the  amplitude  and  direction  of  vibra- 
tion of  an  incident  beam  of  polarized  light. 
This  beam  is  broken  up  by  the  prism  into 
ordinary  and  extraordinary  rays,  of  which  the 
vibrations  are  parallel  to  the  diagonals  AB  and 
CD  respectively.  The  amplitudes  of  these 
vibrations  are  represented  by  the  lines  o  and  e.  The  ampli- 

*  The  plane  of  the  paper  is  a  principal  plane  of  the  crystal,  as  drawn,  and  the 
vibrations  of  the  ordinary  ray  and  of  the  extraordinary  ray  are  in  the  plane  of  the 
paper,  and  perpendicular  thereto  respectively.  Compare  Art.  748. 


o 
Fig.  526. 


132 


ELEMENTS   OF   PHYSICS. 


tude  of  the  extraordinary  ray,  which  passes  through  the  prism, 
is  a  cos  cf>.  Its  intensity  is  to  the  intensity  of  the  incident  beam 
as  the  square  of  its  amplitude  is  to  the  square  of  the  ampli- 
tude of  the  incident  beam.  That  is,  T  \  I  =  #2cos2c/> :  a2,  or 

r=/cos2<£,  (340) 

in  which  /  is  the  intensity  of  the  incident  beam,  and  T  is  the 
intensity  of  the  transmitted  beam.  T  evidently  varies  from  a 
maximum  when  cos$  =  I,  to  zero,  when  cos  <f>  =  o. 

752.  The  polariscope  consists  of  a  Nicol  prism  P  (Fig.  527) 
called  the  polarizer,  which  produces  a  beam  of  polarized  light ; 
and  another  Nicol  prism  A,  called  the  analyzer,  through  which 


Fig.  527. 

this  polarized  beam  passes,  in  whole  or  in  part,  to  the  eye. 
Both  prisms  are  arranged  to  turn  about  the  axis  of  the  instru- 
ment. The  object  which  is  to  be  examined  is  placed  in  the 
polarized  beam  between  the  prisms.  Sometimes  it  is  desired 
to  examine  an  object  in  a  convergent  beam  of  polarized  light 
as  at  cc.  To  this  end  a  lens  L  is  provided  which  gives  a  con- 
vergent beam.  A  second  lens  L'  renders  the  beam  again  par- 
allel before  it  reaches  the  analyzer. 

Two  glass  plates  placed  as  shown  in  Fig.  519,  and  arranged 
to  turn  about  the  polarized  beam,  as  an  axis,  also  constitute  a 
polariscope.  Two  tourmaline  plates  used  as  polarizer  and  ana- 
lyzer are  likewise  sometimes  employed ;  they  form  a  very  con- 
venient and  inexpensive  polariscope. 

753.  Appearance  in  the  polariscope  of  a  thin  plate  of  a  doubly 
refracting  crystal.  —  The  polarized  beam  from  the  polarizing 
Nicol  is  resolved  by  the  crystalline  plate  into  two  beams  (see 
Art.  751).  These  two  beams  (ordinary  and  extraordinary)  pass 
through  the  plate  at  different  velocities,  so  that  one  of  them  is 


POLARIZATION   AND   DOUBLE   REFRACTION. 


133 


retarded  with  respect  to  the  other.  A  component  of  each  of 
these  beams  passes  through  the  analyzer.  Upon  emergence 
these  transmitted  components  are  polarized  in  the  same  plane, 
and  one  of  them  being  retarded  relatively  to  the  other,  they 
interfere.*  Some  wave  lengths  are  thus  strengthened  while 
others  are  weakened,  and  brilliant  color  effects  are  produced. 
This  action  is  shown  beautifully  by  thin  plates  of  mica  and  by 
glass  plates  which  have  been  rendered  doubly  refracting  by 
elastic  strain. 

754.   Rotation  of  plane  of  polarization ;  the  saccharimeter.  — 

Many  substances  have  the  property  of  turning  the  plane  of 
polarization  of  a  transmitted  ray  about  the  ray  as  an  axis. 
The  amount  of  turning  is  proportional  to  the  distance  which 
the  waves  travel  through  the  substance,  and  varies  with  the 
wave  length  of  the  light,  and  with  the  nature  of  the  substance. 
Crystals  of  potassium  chlorate  and  of  quartz,  turpentine,  and 
solutions  of  sugar,  are  examples. 

In  the  case  of  sugar  solutions  the  rotation  of  the  plane  of 
polarization,  for  light  of  a  given  wave  length,  is  proportional  to 
the  distance  traveled  by  the  light  in  the  solution  and  to  the 
strength  of  the  solution  ;  so  that 

a  =  k  •  Im.  (341) 

In  this  equation  a  is  the  angle  (in  degrees)  of  rotation  pro- 
duced when  polarized  light  passes  through  /  cm.  of  a  solution 
containing  m  grams  per  ex.  of  cane  sugar.  The  proportionality 
constant  k,  for  sodium  light,  at  the  ordinary  room  temperatures 

has  the  value  — - —     When  k  is  known,  and  a  and  /  have  been 

•    1-504 
observed,  the  strength  of  the  syrup  may  be  computed  by  means 

of  equation  (341).  A  polariscope  provided  with  a  tube,  to  con- 
tain the  syrup,  and  arranged  for  the  measurement  of  a  is  called 
a  saccharimeter. 

*  Two  beams  of  which  the  planes  of  polarization  are  at  right  angles  do  not  inter- 
fere, inasmuch  as  the  displacements  at  a  point  due  to  the  respective  beams  are  at 
right  angles  to  each  other. 


134 


ELEMENTS    OF   PHYSICS. 


755.  Electro-magnetic  rotation  of  the  plane  of  polarization.  — 
When  polarized  light  is  passed  parallel  to  the  lines  of  force 
through  a  transparent  medium  in  a  magnetic  field,  a  rotation  of 
the  plane  of  polarization  is  produced.  For  example,  carbon 
bisulphide  in  a  tube  surrounded  by  a  coil  of  wire  carrying  cur- 
rent, rotates  the  plane  of  polarization  of  light  which  is  sent 
through  the  tube. 

This  phenomenon  has  been  utilized  by  Crehore  and  Squier, 
in  their  photo-chronograph,*  which  is  used  in  measuring  the 
velocity  of  projectiles.  It  consists  of  a  tube  of  carbon  bisul- 
phide C  surrounded  by  a  coil  of  wire.  This  tube  is  mounted 
between  two  Nicol  prisms  (Fig.  528).  These  are  crossed,  and  so 


Fig.  528. 

long  as  no  current  flows  through  the  coil,  no  light  can  pass.  The 
momentary  currents  which  form  the  chronographic  signals,  and 
which  it  is  desired  to  record,  turn  the  plane  of  polarization 
within  the  tube,  and  flashes  of  light  pass  through  the  second 
Nicol  prism  JVZ.  These  flashes  are  recorded  photographically 
upon  a  rotating  disk  Z>.f 

*  Journal  of  the  United  States  Artillery,  Vol.  VI.,  No.  3. 

t  For  a  fuller  treatment  of  polarization  and  double  refraction  the  student  is 
referred  to  Preston,  Theory  of  Light;  Groth,  Physikalische  Krystallographie,  and  to 
the  original  memoirs  of  Fresnel. 


CHAPTER    XII. 
RADIATION. 

756.  Radiant  heat.  —  The  various  homogeneous  components 
of  the  radiation  from  the  sun,  or  from  any  hot  body  or  luminous 
body,  have  this  common  property ;  namely,  that  they  generate 
heat  in  a  body  which  absorbs  them.  Such  radiation  is  there- 
fore called  radiant  heat.*  The  energy  per  second  streaming 
across  unit  area  at  right  angles  to  the  ray  is  taken  as  the 
physical  measure  of  its  intensity. 

The  radiation  from  a  hot  body,  such  as  the  sun,  is  composed 
of  numerous  homogeneous  components  which  differ  from  one 
another  only  in  wave  length. 

The  range,  as  to  wave  length,  however,  seems  to  be  almost 
infinite.  Rubens  and  Nichols  f  (E.  F.)  have  isolated  and  identi- 
fied homogeneous  components  of  the  radiation  from  hot  zirconia 
of  ^  mm.  wave  length.  These  waves  were  isolated  by  repeated 
reflection  from  fluorite  (compare  footnote  to  Art.  767),  and  the 
wave  length  was  determined  by  the  use  of  a  coarse  diffraction 
grating  made  of  wires. 

The  wide  gap  in  wave  length  between  the  radiation  due  to 
electrical  disturbances  and  the  longest  waves  previously  recog- 
nized in  the  radiation  from  hot  bodies  is  thus  bridged,  and 
homogeneous  radiation  of  every  wave  length  from  Hertz  waves, 
several  meters  or  even  kilometers  long,  down  to  ultra-viplet  rays 
of  less  than  2O-io~~6  cm.  in  wave  length,  is  definitely  known*. 

*  Sometimes  called  radiant  energy.  Radiation,  in  systems  in  equilibrium,  how- 
ever, conforms  to  both  laws  of  thermodynamics,  and  may  more  properly  be  called 
radiant  heat. 

t  Physical  Review,  Vol.  IV.,  1897. 

135 


OF  THK 

IVERSITT 


136  ELEMENTS   OF   PHYSICS. 

Becquerel*  has  described  rays  which  are  emitted  by  the  ura- 
nium salts,  and  which  appear  to  be  of  much  shorter  wave 
length  than  any  of  the  ultra-violet  rays  hitherto  observed. 

757.  Luminous  effects  and  chemical  effects  of  radiant  heat.  - 

Homogeneous  radiation,  of  which  the  wave  length  lies  between 
39-icr6  cm.  and  75-io~6  cm.,  affects  the  optic  nerves  and  pro- 
duces sensations  of  light.  These  limits  are  not  sharply  defined, 
but  vary  greatly  with  different  persons,  with  the  intensity  of 
the  radiation,  and  with  the  degree  of  fatigue  of  the  optic  nerves. 
The  chemical  effect  of  radiation  is  exemplified  in  the  reduc- 
tion of  carbon  dioxide  in  the  growth  of  plants,  in  the  bleaching 
action  of  bright  sunlight,  in  the  action  of  light  upon  photo- 
graphic sensitive  plates,  and  so  on.  The  intensity  of  this  chemi- 
cal action  varies  greatly  with  the  wave  length.  The  particular 
wave  length  for  which  the  chemical  action  is  greatest  (for  a 
given  energy  intensity  of  the  radiation)  varies  with  the  sub- 
stance upon  which  the  action  is  exerted.  For  the  salts  of  silver 
used  in  photography  the  green,  blue,  and  violet  rays  are  most 
active,  while  the  extreme  red  rays  are  almost  wholly  inactive. 

758.  Prevost's  principle  of  exchanges.  —  A  number  of  bodies 
in  an  inclosed  region  exchange  heat   by  radiation  until  they 
reach  uniform  temperature.     The  whole  system  is  then  in  ther- 
mal equilibrium.     Radiation  persists  in  a  region  after  the  region 
has  settled  to  thermal  equilibrium.     It  is  almost  necessary  to 
suppose  that  the  molecular  commotion  in  the  various  bodies 
continues  to  produce  waves  even  after  all  the  bodies  in  a  region 
have  reached  uniform  temperature.     Each  body  then  gives  off 
as  much  radiant  heat  as  it  receives.     If  the  temperature  of  one 
body  is  low,  it  radiates  less  than  it  receives,  and  its  tempera- 
ture rises ;  if  its  temperature  is  high,  it  radiates  more  than  it 
receives,   and   its    temperature   falls.      These   facts  were   first 
pointed  out  by  Prevost.     They  comprise  what  is  known  as  the 
principle  of  exchanges. 

*  Comptes  Rendus,  122  (1896). 


RADIATION.  137 

759.  The  law  of  normal  radiation. — The  radiation   coming 
from  a  body,  at  a  given  temperature,  is  of  three  distinct  parts : 
(i)  rays  emitted  by  the  body ;  (2)  rays  reflected  from  its  sur- 
face ;  (3)  rays  transmitted  by  the  body  from  radiating  surfaces 
behind  it.     These,  taken  together,  constitute  what  is  called  the 
total  radiation.     A  system  or  region  which  neither  gains  nor 
loses  heat  is  called  a  closed  system  or  region.     The  law  of  radia- 
tion for  such  a  system  may  be  stated  as  follows  : 

In  a  closed  region  at  a  given  uniform  temperature,  the  total 
radiation  from  any  body,  whatever  its  nature,  is  of  definite  com- 
position. In  other  words,  each  of  the  various  homogeneous  com- 
ponents of  the  total  radiation  is  of  definite  intensity.* 

Proof.  —  Experience  shows  that  a  number  of  bodies  come  to 
a  state  of  thermal  equilibrium  when  left  to  themselves  in  a  closed 
region ;  and  that  this  state  is  not  disturbed  in  any  way  when  a 
foreign  body  at  the  same  temperature  is  introduced  into  the 
region. 

Remark. — The  radiation  in  a  closed  region  at  a  given  uniform 
temperature  is  called  the  normal  radiation  for  that  temperature. 

The  term  normal  applies  both  to  composition  and  to  inten- 
sity. Both  the  radiation  impinging  upon  a  body  and  that  sent 
out  from  it  in  such  a  closed  region,  are  normal. 

760.  Proposition.     Emission  and  absorption  are  equal.  —  Con- 
sider a  body  in  a  region  in  thermal  equilibrium.     The  total 
radiation  falling  upon  the  body  and  the  total  radiation  from  it 
are  normal.     A  portion  of  the  latter  is  reflected.     Let  this  por- 
tion be  removed,  and  an  equal  amount  be  subtracted  from  the 
incident  radiation.     Another  portion  of  the  total  radiation  from 
the  body  is  transmitted ;  imagine  this  to  be  removed  and  a  cor- 
responding amount  to  be  subtracted  from  the  incident  radiation. 

*  In  1866  Kirchhoff  published  a  theorem,  known  as  KirchhoflPs  law,  viz. :  for  a 
given  temperature,  the  relation  between  emissive  power  and  absorbing  power  is  for  all 
bodies  the  same.  The  import  of  this  theorem  is  in  accordance  with  the  statement 
given  above. 


138 


ELEMENTS   OF   PHYSICS. 


The  remaining  portion  of  the  incident  radiation  is  absorbed  by 
the  body ;  the  remaining  portion  of  the  radiation  from  the  body 
is  emitted,  and  these  two  portions  are  equal. 

761.  Selective  emission,  reflection  and  transmission.  —  In  the 

case  of  many  substances,  the  total  radiation  for  certain  wave 
lengths  consists  almost  entirely  of  emitted  light,  while  for  other 
wave  lengths  the  emission  is  very  slight,  and  the  total  radiation 
is  made  up  chiefly  of  transmitted  or  reflected  light.  Gases  fur- 
nish the  most  striking  example.  Such  substances  are  said  to 
exhibit  selective  emission. 

When  the  total  radiation  for  certain  wave  lengths  contains 
a  greater  proportion  of  transmitted  (or  reflected)  light,  than  is 
the  case  for  other  wave  lengths,  the  substance  is  said  to  exhibit 
selective  transmission  (or  reflection}. 

762.  Selective   absorption.  —  It   is  frequently  convenient   to 
express  the  behavior  of  a  body  by  reference  to  its  absorbing 
power  for  radiation.     Since  emission  and  absorption  are  always 
equal,  wave  length  for  wave  length,  all  bodies  which  show  selec- 
tive emission  show  selective  absorption  also. 

763.  Ideal  cases  of  selective  action.  —  There  are  four  ideal 
cases,  the  consideration  of  which  will  help  towards  the  under- 
standing of  the  behavior  of  various  substances  which  approxi- 
mate to  them.  i 

(a)  Bodies  which  do  not  reflect  perceptibly.  —  Such  bodies  do 
not  transmit  (that  is,  they  do  absorb)  those  wave  lengths  which 
they  emit  in  excess.     In  this  case  the  transmitted  and  emitted 
radiations  are  complementary. 

(b)  Bodies  which  are  opaque  (i.e.,  which  do  not  transmit).  — 
Such  bodies  emit  radiations  which  are  complementary  to  the 
radiations  which  they  reflect. 

(c)  Bodies   which  do   not   radiate  perceptibly.  —  Such   bodies 
reflect   best  those  wave   lengths  which   they  do  not  transmit. 


RADIATION.  139 

That  is  to  say,  the  reflected  radiations  and  transmitted  radia- 
tions are  complementary. 

(d)  Opaque  bodies  which  do  not  reflect  (black  bodies).  —  Such 
bodies  emit  normal  radiation. 

These  peculiarities  persist  even  when  the  incident  radiation 
is  not  the  normal  radiation  corresponding  to  the  temperature  of 
the  body,  although  in  such  a  case  the  two  portions  into  which 
the  radiations  are  divided  are  not  to  be  thought  of  as  comple- 
mentary in  the  complete  sense  in  which  this  term  is  used  above. 

764.  Existing  cases  of  selective  action.  —  (i)  Gases  do  not  re- 
flect perceptibly,  and  when  hot  they  radiate  those  wave  lengths 
in  excess  which  they  absorb  excessively  when  cool  (Arts.  699 
and  700).  Ruby  glass,  which  is  red  by  transmitted  light,  and 
which  shows  no  marked  selective  action  by  reflection,  is  green 
when  it  is  heated  to  incandescence. 

(2)  Metallic  copper  is  sensibly  opaque.     It  is  red  by  reflected 
light,  but  when  incandescent,  it  is  green. 

(3)  Fuchsine,  of  the  emitting  power  of  which  nothing  defi- 
nite is  known,  transmits  red  and  violet  light  and  reflects  green 
light   almost    completely.     The   approximately   complementary 
character   of    the   transmitted    and    reflected    radiation    shows 
indeed  that  the  emitting  power  of  fuchsine  is  relatively  small. 
In  dilute  solution  its  behavior  is  somewhat  different. 

Gold  is  yellow  by  reflected  light,  and  gold  leaf  is  beautifully 
green  by  transmitted  light.  It  follows  that  an  extremely  thin 
sheet  of  gold  would  not  exhibit  selective  emission  when  hot. 
A  thick  sheet  (opaque)  appears  greenish  when  incandescent, 
but  the  effect  is  not  so  marked  as  in  the  case  of  copper,  because 
the  color  of  the  metal  by  reflected  light  is  not  so  ruddy. 

Remark. — An  opaque  body  which  reflects  all  wave  lengths 
well  radiates  very  little,  and  an  opaque  body  which  reflects  very 
little  radiates  well.  Thus  dead  black  steam  pipes  radiate  much 
better  than  brightly  polished  ones.  Water  cools  much  more 
slowly  in  a  polished  metal  vessel  than  in  a  blackened  one. 


I40  ELEMENTS   OF   PHYSICS. 

765.  Black  bodies.  —  An    opaque  body  which   reflects  very 
little  of  the  radiation  which  falls  upon  it  is  said  to  be  black. 
Such  bodies,  when  heated  to  a  given  temperature,  emit  very 
nearly  the  normal  radiation  for  that  temperature.     Amorphous 
carbon,  graphitic  carbon,  and  the  black  oxide  of  iron  are  exam- 
ples of  nearly  black  bodies. 

Perfectly  black  substances  do  not  occur.  A  perfectly  black 
body  would  be  an  opaque  body  which  reflected  no  portion  of 
the  rays  falling  upon  it.  A  small  hole  in  the  opaque  wall  of  a 
large  closed  chamber  is  perfectly  black,  for  only  an  infinitesimal 
portion  of  the  rays  which  enter  find  their  way  out  again.  If 
such  a  chamber  be  maintained  at  a  uniform  temperature,  the 
radiation  emitted  from  the  hole  will  be  the  same  as  the  radiation 
within,  which  is  normal. 

The  appearance  of  a  large,  uniformly  heated  tile  kiln,  as  seen 
through  a  hole  in  the  wall,  is  very  striking.  Even  though  the 
interior  be  partially  free  from  tiles,  nothing  can  be  seen  but  a 
flood  of  soft  yellow  light.  The  radiation  reflected,  transmitted 
(if  any),  and  emitted  by  a  tile,  or  by  a  piece  of  the  air  for  that 
matter,  is  normal,  so  that  identical  radiations  reach  the  eye 
from  every  portion  of  the  interior.  If  the  peep  hole  is  large 
enough  to  cool  the  adjacent  tiles,  they  become  faintly  visible ; 
or  if  a  beam  of  sunlight  (it  must  be  light  from  something  hotter 
than  the  kiln)  is  reflected  into  the  opening,  the  tiles  become 
visible,  as  if  they  were  in  a  dark  chamber. 

766.  White  bodies.  —  A  body  which  reflects  (approximately) 
the  same  proportion  of  the  various  homogeneous  components  of 
the  radiation  which  falls  upon  it  is  called  a  white  body.     The 
light  reflected  by  a  white  body  is  of  the  same  composition  as 
the  incident  light.     Thus  white  paper  is  red  in  red  light,  green 
in  green  light,  etc.     A  perfectly  white  body  would  be  one  reflect- 
ing the  whole  of  the  incident  radiation,  as  opposed  to  a  perfectly 
black  body  which  absorbs  the  whole.     No  substance  is  perfectly 
white.     There  is  no  substance,  even,  which  reflects  exactly  the 


RADIATION.  I4I 

same  proportion  of  each  homogeneous  component  *"of  the  inci- 
dent radiation.  Most  white  powders  —  for  example,  sugar, 
magnesium  oxide,  etc.  —  reflect  the  longer  wave  lengths  in 
excess,  and  have  a  yellowish  tint.  This  is  counteracted  by  the 
admixture  of  a  small  amount  of  a  powder,  e.g.  ultramarine  blue, 
which  reflects  the  shorter  wave  lengths  in  excess. 

Substances,  like  glass  and  ice,  which  transmit  (and  reflect) 
every  part  of  the  visible  spectrum  equally  well  (approximately), 
are  dazzling  white  when  powdered.  This  is  exemplified  in  the 
whiteness  of  snow.  Nearly  the  whole  of  the  light  falling  upon 
a  sheet  of  snow  is  reflected  (diffusely),  because  of  the  repeated 
reflections  by  the  successive  particles  as  they  are  reached  by 
the  light  as  it  penetrates  deeper  and  deeper  into  the  snow. 

767.  Surface   color.*  —  A    substance    which    shows    marked 
selective   reflection   is   said   to   have   surface   color.     All   such 
substances  appear  different  in  color  by  reflected  and  by  trans- 
mitted light.     Thus  gold  is  yellow  by  reflected  light,  and  gold- 
leaf  is  beautifully  green    by  transmitted   light.      The   aniline 
dyes,  especially  when   concentrated,  are  different  in   color  by 
reflected  and  by  transmitted  light.     Red  wine  is  of  a  beautiful 
bronze  color  by  reflected  light. 

768.  Absorption   color.  —  Most   colored   substances,  such   as 
colored   glass,  colored  solutions,  etc.,  show  color,  perceptibly, 
by  transmitted   light    only.     Many    colored    substances    which 

*  The  electro-magnetic  theory  of  light  shows  :  (i)  That  a  perfect  electrical  con- 
ductor would  totally  reflect  all  radiations  falling  upon  it;  (2)  That  a  substance  of 
which  the  structural  elements  have  a  proper  period  of  undamped  electrical  vibration 
would  totally  reflect  wave  trains  of  that  particular  period;  and  (3)  That  in  pro- 
portion as  these  proper  vibrations  are  damped  the  substance  would  reflect  less  and 
absorb  more  of  the  trains  having  its  proper  period.  An  example  of  the  first  case 
is  furnished  by  metals,  which,  especially  for  the  longer  wave  lengths,  give  almost 
complete  reflection.  Such  reflection  is  called  metallic  reflection. 

Many  substances  such  as  fluorite  and  fuchsine  reflect  certain  wave  lengths  almost 
completely.  Repeated  reflection  from  fluorite,  for  example,  isolates  the  wave  train 
which  is  most  completely  reflected.  Reflection  of  this  kind,  for  want  of  a  better 
name,  is  called  anomalous  reflection  (surface  color).  It  might  well  be  called  reso- 
nant reflection. 


142 


ELEMENTS    OF   PHYSICS. 


appear  colored  by  reflected  light,  such  as  the  pigments  used 
in  painting,  really  owe  their  color  to  selective  transmission  or 
absorption.  The  light  falling  upon  their  grains  is  partially 
transmitted,  becomes  colored,  and  is  reflected  by  numerous 
foreign  particles  and  by  breaks  in  the  continuity  of  the  grains. 
This  is  shown  by  the  fact  that  a  mixture  of  two  pigments 
reflects  only  those  wave  lengths  which  are  reflected  by  both 
pigments  unmixed,  just  as  a  pair  of  colored  glasses  transmits 
only  those  wave  lengths  which  can  pass  through  both.  If  the 
pigments  were  colored  mainly  by  true  selective  reflection,  each 
isolated  grain  of  each  pigment  would  reflect  independently,  and 
all  the  wave  lengths  reflected  by  each  pigment  would  be  found 
in  the  light  reflected  by  the  mixture. 

769.  Methods  of  measuring  radiant  heat.  —  The  study  of  the 
composition  of  radiation  is  a  matter  of  great  difficulty.  In 
the  experimental  determination  of  the  distribution  of  energy  in 
the  spectrum  of  a  glowing  body,  the  selective  transmission  of  the 
prisms  and  lenses  of  the  spectrometer  introduces  uncertainties 
which  cannot  be  wholly  eliminated  ;  and  if  a  concave  diffraction 
grating  is  used,  it  is  impossible  to  know  the  details  of  the  grating 
with  sufficient  accuracy  to  be  able  to  calculate  the  intensity  of 
a  homogeneous  component  of  the  radiation  from  the  observed 
intensity  of  the  corresponding  part  of  the  spectrum. 

The  instruments  used  in  such  work  are  the  thermopile,  the 
bolometer,  and  the  radiometer. 

The  thermopile  has  been  described  in  Vol.  II.  (Art.  558). 

The  bolometer,  invented  by  Langley  *  in  1880,  consists  of  a 
Wheatstone  bridge,  one  of  the  arms  of  which  is  made  of  a  thin 
strip  of  blackened  metal.  This  is  exposed  to  the  rays,  the 
intensity  of  which  is  to  be  indicated.  The  radiation  causes  a 
rise  in  temperature  and  a  consequent  increase  in  the  resistance 
of  the  metal  strip.  This  affects  the  balance  of  the  bridge,  and 
a  sensitive  galvanometer  in  the  bridge  circuit  shows  a  deflection. 

*  See  Transactions  of  the  National  Academy  of  Sciences,  Vol.  V. 


RADIATION. 


143 


jf 


Y 


The  radiometer  used  in  the  measurement  of  radiation  differs 
essentially  from  the  radiometer  of  Crookes.  It  was  invented 
by  Ernest  Nichols*  in  1896.  It  consists 
of  two  similar  thin  vanes  of  blackened 
mica  (aa)  attached  to  a  horizontal  arm,  and 
suspended  in  a  high  vacuum  by  means  of 
a  fine  quartz  fiber  (Fig.  529).  The  radia- 
tion to  be  indicated  falls  upon  one  of 
these  vanes  and  warms  it  slightly.  This 
causes  the  few  remaining  molecules  of  air 
to  rebound  with  increased  velocity  from 
the  blackened  face.  The  reaction  pushes 
the  vane  backwards,  and  turns  the  arm 
about  the  fiber  as  an  axis.  The  deflec- 
tion is  observed  by  means  of  a  telescope 
and  scale.  The  sensitiveness  of  this  in- 
strument is  such  that  the  rays  from  a 
candle  at  a  distance  of  450  meters  (nearly  one-third  of  a  mile) 
produce  a  noticeable  deflection. 

770.  The  comparison  of  various  sources  of  light  by  means  of 
the  bolometer.  —  Langley's  measurements  of  the  spectra  of 
various  sources  of  radiation  afford  good  illustrations  of  the  use 
of  the  bolometer. 


aHa 

1 

— 

- 

1 

Fig.  529. 


600- 
400 

200- 


V  R  l.O/i  2.0//  3.0/i 

Fig.  530. 

The   ordinates  of  the  curves  in  Fig.   530  show  the  relative 
intensities  of  the  different  homogeneous   components   in    sun- 

*  See  Physical  Review,  Vol.  IV.,  p.  297. 


I44  ELEMENTS   OF   PHYSICS. 

light,  gaslight,  and  in  the  light  of  the  electric  arc  lamp.  The 
deep  notches  in  the  curve  of  sunlight  are  absorption  bands,  due 
to  relatively  cool  vapors  around  the  sun  and  in  the  earth's 
atmosphere.  The  dotted  line  shows  approximately  what  the 
radiation  from  the  sun  would  be  were  it  not  for  this  selective 
absorption. 

771.  Coefficients  of  absorption.  —  Let  /  be  the  intensity  of  a  homogeneous 
beam  of  light  as  it  enters  a  plate  of  indefinite  thickness.  Let  i  be  the  intensity 
of  the  beam  after  having  penetrated  to  a  distance  x  into  the  plate  ;  and  let  Az 
be  the  further  decrease  in  intensity  of  the  beam  as  it  penetrates  to  an  addi- 
tional distance  kx  into  the  plate.  The  portion  Az  of  the  beam  which  is 
absprbed  is,  by  experiment,  sensibly  proportional  to  i  and  to  A^r,  so  that 
A*'  —  —  ki  .  Ar,  or 


In  this  equation  k  is  the  proportionality  factor.  The  negative  sign  is  chosen 
inasmuch  as  A/  is  a  decrement  of  intensity.  By  integration,  equation  (i)  gives 
Loge  i  =  —  kx  +  a  constant,  or 


in  which  C  is  an  undetermined  constant.  When  x=o,  thenz'  =  /.  These 
values  substituted  in  (ii)  give  C  —  /,  so  that  equation  (ii)  becomes 

i  =  I*'**,  (342) 

in  which  /  is  the  intensity  of  a  beam  of  homogeneous  light  as  it  enters  an 
absorbing  substance,  i  is  its  intensity  when  it  has  penetrated  to  a  depth  x,  e  is 
the  Naperian  base,  and  k  is  a  constant  called  the  coefficient  of  absorption,  of 
the  substance  for  the  particular  kind  of  homogeneous  light. 

For  substances  which  exhibit  selective  absorption,  the  value  of  k  depends 
upon  the  wave  length  of  the  homogeneous  beam. 

The  law  of  decrease  of  intensity  of  a  beam,  as  expressed  by  equation  (342), 
is  as  follows  : 

Of  the  beam  which  remains  unabsorbed  the  same  fractional  part  is  absorbed 
in  each  successive  layer  of  the  substance. 

772.  Helmholtz's  theory  of  dispersion.  —  Consider  a  stretched  rubber 
heavier  at  one  end  than  at  the  other,  AB  (Fig.  531),  along  which  trans- 
verse waves  may  be  sent  by  moving  the  end  A  back  and  forth.  Let  c,  c,  c, 
etc.,  be  equal  weights  suspended  from  the  tube  by  similar  spiral  springs,  so 
that  each  of  these  weights  has  the  same  proper  period  of  vibration  T.  Wave 
trains  of  all  wave  lengths  will  move  along  AD  at  the  same  velocity  v, 
and  ignoring  the  action  of  the  weights,  along  DB,  the  heavier  end,  at  a 


RADIATION. 


velocity  -  v.  The  action  of  the  weights  is  to  cause  /u,  to  vary  with  the  peri- 
odic time  T  of  the  wave  train.  Helmholtz's  theory  of  dispersion  is  represented 
mechanically  by  this  arrangement.  Helmholtz  showed  *  that  the  velocity  of 


o 


C    C    C  ETC. 

Fig.  531. 

a  wave  train  along  DB  is  increased  (/x  decreased)  by  the  action  of  the  weights, 
so  long  as  r  is  less  than  T,  and  decreased  (/x,  increased)  so  long  as  T  is  greater 
than  T,  and  that  the  effect  of  the  weights  becomes  greater  as  T  approaches  T. 
When  T  =  T,  an  incident  wave  train  along  AD  is  almost  totally  reflected 
from  A,  especially  if  the  motion  of  the  weights  is  frictionless ;  and,  in  any 
case,  the  action  along  DB  when  T  =  T  is  not  of  the  nature  of  a  wave  train  at  all. 

The  ordinates  of  the  curve  (Fig.  532)  represent  the  values  of  u  fvelocity  ™DB\ 

\  velocity  in  AD  I 

for  different  values  of  the  period  r  of  the  incident 
wave  train.  The  abscissa  of  the  ordinate  T  rep- 
resents the  periodic  time  of  the  suspended 
weights. 

Examples.  —  The  Helmholtz  theory  shows 
that  the  refractive  index  of  a  substance  is  very 
greatly  increased  below  an  absorption  band 
(r>Z>),  and  greatly  decreased  above  an  ab- 
sorption band  (T  <  Z1),  and  that  wave  trains  for 
which  r  =  T  are  almost  wholly  reflected.  This, 
in  fact,  is  found  to  be  the  case  for  those  sub- 
stances which  exhibit  powerful  selective  action 
and  show  surface  color.  This  wide  variation  of  the  refractive  index  in  the 
neighborhood  of  an  absorption  band  has  been  called  anomalous  dispersion, 
or,  better,  resonant  dispersion. 


T 
Fig.  532, 


773.    Example  of    resonant    dispersion.  — 

The  ordinates  of  the  dotted  curve  in  Fig. 
533  show  the  refractive  index  of  a  solution 
of  fuchsine  for  rays  of  various  wave  lengths. 
The  band  AB  shows  the  actual  appearance  of 
a  narrow  spectrum  CD  when  viewed  through 
a  prism  containing  a  solution  of  fuchsine  held 
in  such  a  way  as  to  deflect  the  spectrum  CD  laterally. 


Cl 


ID 


Fig.  533. 


*  See  Berliner  Berichte,  1874,  p.  667. 
2'(i),  p.  674. 
L 


Also  Winkelmann,  Handbuch  der  Physik, 


I46  ELEMENTS   OF  PHYSICS. 

774.  Phosphorescence  and  fluorescence.  —  Some  substances, 
while  undergoing  slow  chemical  action,  emit  radiation  of  the 
shorter  wave  lengths  greatly  in  excess  of  the  normal  amount  cor- 
responding to  the  temperature  of  the  substance.  Thus  phos- 
phorus oxidizes  slowly  when  it  is  exposed  to  the  air  at  a  low 
temperature,  and  emits  a  pale  white  light.  This  action  is  some- 
times called  phosphorescence.  The  term  is,  however,  more 
generally  applied  to  the  phenomenon  described  below. 

Many  substances  —  for  example,  calcium  sulphide  —  glow  for 
a  time  in  the  dark,  after  being  exposed  to  intense  radiation. 
Such  substances  are  said  to  be  phosphorescent.  The  light  given 
off  by  a  phosphorescent  substance  is  generally  of  greater  wave 
length  than  the  light  which  is  needed  to  bring  the  substance 
to  phosphorescence.  A  very  remarkable  exception  is  the  case 
of  the  salts  of  uranium.  Becquerel  (See  Art.  756)  has  found 
that  these  salts  send  out  rays  of  extremely  short  wave  lengths 
for  a  very  long  time  after  being  exposed  to  sunlight.  These 
rays  (Becquerel  rays)  are  of  much  shorter  wave  length  than  any 
hitherto  recognized  in  the  ultra-violet  of  the  solar  spectrum. 

Some  substances  —  as  sulphate  of  quinine,  kerosene,  and 
uranium  glass  —  emit  light  of  medium  wave  length  when 
exposed  to  radiation  of  very  short  wave  length.  Such  sub- 
stances are  said  to  be  fluorescent  or  luminescent.  The  Rontgen 
rays  produce  strong  luminescence  in  some  substances.  (See 
Vol.  II.,  Art.  498.) 


CHAPTER   XIII. 


LOUDNESS,   PITCH,   AND   TIMBRE. 

775.  The  vibration  of  a  particle.  Simple  and  compound 
vibrations.  —  When  a  particle  moves  to  and  fro  along  a  straight 
line,  performing  simple  harmonic  *  motion,  its  vibrations  are  said 
to  be  simple. 

When  the  motion  of  a  particle  is  periodic,  but  not  simply 
harmonic,  its  vibrations  are  said  to  be  compound. 
A 


B 

Fig.  534  a. 

Graphical  representation  of  simple  and  compound  vibrations.  — 
Consider  a  point  /  (Fig.  534  a)  vibrating  up  and  down  along 
the  line  AB.  Imagine  the  paper  to  move  with  uniform  velocity 
to  the  right,  then  the  point  p  will  trace  a  line  cc.  If  the 
vibrations  of  /  are  simple,  this  curve  cc  will  be  a  curve  of  sines. 
If  the  vibrations  of  /  are  compound,  the  curve  cc  will  be  a 
periodic  curve,  i.e.  each  section  of  it  exactly  similar  to  every 
other  section,  but  not  a  curve  of  sines.  The  curve  in 
Fig.  534  b,  which  shows  the  vibratory  motion  of  a  point  of  a  violin 
string,  is  such  a  compound  curve. 

The  number  of  (complete)  vibrations  per  second  of  a  particle 
is  called  the  frequency  of  the  vibrations. 

*  See  Vol.  I.,  Art.  59. 


I48  ELEMENTS   OF   PHYSICS. 

The  time  of  one  complete  vibration  is  called  the  period  of 
the  vibrations.  (See  Art.  59,  Vol.  I.,  for  definitions  of  ampli- 
tttde,  phase,  and  phase  difference^ 


Fig.  534  b. 

Remark. — The  conception  of  simple  and  compound  vibra- 
tions of  a  particle  may  be  applied  to  the  vibrations  of  a  body 
which  moves  up  and  down  or  to  and  fro  sensibly  as  a  whole. 
For  example,  the  end  of  the  prong  of  a  tuning  fork  performs 
simple  vibrations  ;  a  portion  of  the  sounding  board  of  a  piano 
performs  compound  vibrations. 

776.  Fourier's  theorem  applied  to  the  vibrations  of  a  particle. 

—  Any  periodic  vibration  of  frequency  n  is  equivalent  to  a 
number  of  superposed  simple  vibrations  of  which  the  respective 
frequencies  are  n,  2/2,  3/2,  4/2,  and  so  on,  and  of  which  the 
respective  amplitudes  are  determinate.  It  is  for  this  reason 
that  such  vibrations  are  called  compound  vibrations.  The  state- 
ment here  given  of  Fourier's  theorem  is  identical,  from  the 
mathematical  point  of  view,  with  the  statement  of  Art.  624, 
referring  to  simple  and  compound  wave  trains. 

777.  Musical  tones  defined.  —  When  a  simple  or  compound 
wave  train  from  a  vibrating  body  falls  upon  the  ear,  the  result 
is  a  sensation  of  sound.     It  is  necessary  to  the  production  of 
such  a  sensation,  that  the  frequency  of  some  of  the  components 
of  the  wave  train  lie  within  certain  limits  called  the  limits  of 
audibility.     (See  further,  Art.  780.) 


• 

OP  THE 

{    -  -SRSITY 

LOUDNESS,    PITCH,   AND    TIMBRE. 

When  a  wave  train  is  simple,  or  when,  although  compound, 
it  contains  some  component  which  is  so  much  stronger  than 
the  others  as  to  hold  the  attention  of  the  hearer,  the  result  is 
a  musical  tone.  A  simple  tone  is  one  produced  by  a  simple 
wave  train.  A  compound  tone  is  one  produced  by  a  compound 
wave  train. 

778.  Noises.  —  Sound  sensations  not  falling  under  the  defini- 
tion of  musical  tones,  or  not  consisting  of  some  simple  arrange- 
ment or  combination  of  musical  tones  such  that  the  hearer 
can  distinguish  the  various  parts  and  recognize  their  relations 
to  one  another,  may  be  classified  as  noises.  The  distinction 
between  musical  tones  and  noises  is  not  a  definite  one. 

Rattling  noises  are  due  to  irregular  successions  of  sharp 
clicks.  Hissing  and  roaring  noises  are  due  to  complex  and 
rapidly  varying  combinations  of  tones.  In  the  case  of  hissing 
noises  the  tones,  which  may  be  few  in  number,  are  of  very  high 
pitch,  and  in  the  case  of  roaring  noises  the  tones  are  of  lower 
pitch.  All  manner  of  combinations  of  rattling,  roaring,  and 
hissing  noises  occur,  from  those  combinations  of  musical  tones 
which  begin  to  be  so  complicated  that  a  hearer  cannot  distin- 
guish the  various  parts  and  recognize  their  relations  to  one 
another,  to  the  extremely  complex  sounds  from  waterfalls,  trains, 
and  busy  streets.  The  prevalence  in  a  roar  of  a  dominant  tone 
of  medium  pitch  is  apt  to  engage  the  attention  and  leave  the 
accompanying  noise  to  a  great  extent  unnoticed.  Tones  of 
higher  pitch  than  those  used  in  music  (hissing)  engage  the 
attention,  if  they  are  at  all  prominent,  and  give  a  noisy  charac- 
ter to  the  sound. 

Musical  tones  are  generally  accompanied  by  characteristic 
noises.  Thus,  the  whispering  noises  of  the  breath,  and  the 
sounds  of  the  consonants  used  in  articulation,  accompany  the 
musical  tones  of  the  singer ;  and  the  faint  noises  produced  by 
the  fingers,  keys,  and  pedals  always  accompany  piano  music. 
Many  noises,  on  the  other  hand,  are  accompanied  by  character- 


ISO 


ELEMENTS   OF   PHYSICS. 


istic  musical  tones.  A  light  blow  upon  a  floor,  or  upon  a  piece 
of  furniture,  produces  a  faint  musical  tone,  of  short  duration, 
which  is  often  prominent  enough  to  be  easily  distinguishable. 

779.  Loudness. — That  quality  of  sound  which  depends  upon 
the  intensity  of  the  sensation  is  called  loudness.     The  loudness 
of  a  given  tone,  in  so  far  as  it  is  not  affected  by  fatigue  of  the 
organs  of  the  ear,  and  by  attention,  depends  upon  the  energy 
of  the  vibrations  which  produce  it.     Tones  of  medium  or  high 
frequency  are  very  much  louder,  for  the  same  energy  of  vibra- 
tion, than  tones  of  low  frequency. 

780.  Pitch. — That  quality  of  a  musical  tone  which  depends 
upon  the  frequency  of  vibrations  of  the  wave  train  is  called 
pitch.     The  pitch  of  a  tone  is  high  or  low  according  as  the 
frequency  is  great  or  small. 

"^Simple  vibrations  of  lower  frequency  than  about  34  (com- 
pleteV  vibrations  per  second  are  not  heard  as  a  musical  tone, 
and  wh^n  the  frequency  exceeds  35,000  or  40,000  per  second 
the  tone  becomes  inaudible.  These  limits  vary  greatly  for 
different  persons. 

781.  The  measurement  of  pitch.     Direct  and  indirect  methods. 
—  The  direct  measurement  of   pitch   consists   in   counting  the 
number  of  vibrations  per  second,  in  the  production  of  a  given 
musical  tone.     The  instrument  commonly  used  for  this  purpose 
is  the  siren. 

The  siren  consists  of  a  circular  metal  disk  (Fig.  535)  mounted 
upon  a  shaft  upon  which  a  screw  thread  is  cut.     This  screw 
thread  engages  a  gear  wheel  which  actuates 
a  device  for  counting  the  revolutions  of  the 
disk.      The  disk  has  one  or  more  rows  of 
holes.     These   rows  are  circular,  with  their 
Fig.  535.  centers  at  the  axis  of  the  shaft.     This  disk 

rotates  very  near  the  wall  of  a  chamber  containing  air  under 
pressure,  and  the  holes  in  the  disk  come  before  apertures  in 


LOUDNESS,   PITCH,   AND   TIMBRE. 


AIR-CHAMBER 

/ 


this  wall  in  rapid  succession.  (See  Fig.  536.)  The  puffs  of 
air  thus  produced  blend  into  a  musical  note,  the  pitch  of  which 
is  known  from  the  observed  speed  of  the 
disk  and  the  number  of  holes  in  the  row. 
The  disk  is  sometimes  driven  by  an 
electric  motor  and  sometimes  by  the  ac- 
tion of  the  issuing  air.  In  the  latter 
case,  the  holes  in  the  disk  are  inclined 
like  the  vanes  of  a  windmill.  Fis-  536. 

Another  direct  method  consists  in  recording  upon  the  cylin- 
der of  a  rapidly  moving  chronograph  the  vibrations  of  the 
sounding  body  to  the  oscillation  of  which  the  tone  to  be  meas- 
ured is  due.  Sometimes  the  record  is  made  by  means,  of  a 
stylus  attached  to  the  vibrating  body  and  tracing  a  curve  upon 
a  smoked  surface.  Sometimes  a  small  mirror  is  attached  to  the 
vibrating  body  and  a  beam  of  light  is  thrown  from  it  upon  a 
sensitized  surface.  When  the  wave  train  is  simple,  the  tracing 
is  sinuous.  Figure  537  is  a  facsimile  of  such  a  tracing.  The 
pitch  is  determined  by  counting  the  number  of  undulations 
recorded  in  a  second  of  time. 


Fig.  537. 

The  indirect  measurement  of  pitch  consists  simply  in  compar- 
ing the  pitch  of  one  sounding  body  with-  that  of  another,  which 
is  taken  as  a  standard.  The  standard  is  commonly  a  tuning 
fork.  (See  Art.  795.)  Where  the  tones  to  be  compared  are 
nearly  of  the  same  pitch,  the  method  of  beats  is  frequently 
employed.  •  This  method  depends  upon  the  fact,  which  is  more 
fully  discussed  in  Art.  806,  that  when  two  wave  trains,  nearly 
alike  in  frequency,  reach  the  ear  simultaneously  they  are  alter- 
nately in  the  same  phase  (and  reinforce  each  other)  and  in 
opposite  phase  (and  annul  each  other).  The  result  is  a  series 
of  pulsations  which  grow  slower  as  the  two  wave  trains  approach 


152 


ELEMENTS    OF   PHYSICS. 


in  frequency  and  disappear  altogether  when  complete  unison  is 
attained. 

Another  method  of  comparing  vibrations  is  by  means  of 
Lissajous  figiires.  It  may  be  used  whenever  the  two  frequen- 
cies are  in  a  simple  ratio  such  as  i  :  i ;  1:2;  2:3,  etc. 

To  the  two  vibrating  bodies  mirrors  are  attached.  The 
adjustment  is  such  that  the  planes  of  the  two  vibrations  are 
at  right  angles.  If  a  beam  of  light  be  reflected  from  one 
mirror  to  the  other  and  thence  to  a  screen  or  into  the  eyepiece 
of  a  telescope,  the  spot  of  light  thus  formed  moves  in  a  closed 
curve.  The  pattern  thus  produced  is  called  a  Lissajous  figure. 
It  is  that  which  naturally  arises  from  the  combination  of  the 
two  rectilinear  motions  imparted  to  the  beam  by  the  individual 
vibrations.  Figure  538  shows  the  Lissajous  figures  correspond- 
ing to  the  ratios  i  :  i  ;  1:2;  1:3  and  2  :  3.  Each  figure  is 
shown  in  four  phases. 


I  :  I 


2:  3 


Fig.  538. 

Both  of  the  indirect  methods  described  above  are  used  in 
tuning  instruments.  The  vibration  microscope  of  von  Helm- 
holtz,  an  apparatus  for  the  accurate  adjustment  of  tuning  forks, 
is  based  upon  the  method  of  Lissajous'  figures. 

782.  Timbre.  — The  sound  sensation  produced  by  a  compound 
vibration,  i.e.  a  compound  tone,  may  be  regarded  as  composed 
of  a  series  of  simple  tones  corresponding  to  the  various  simple 


LOUDNESS,   PITCH,   AND   TIMBRE.  ^3 

vibrations  which  enter  into  the  composition  of  the  compound 
vibration.  Such  a  sound  is  usually  perceived  as  a  whole,  even 
by  a  practiced  ear,  and  may  be  called,  following  the  German,  a 
clang. 

A  very  little  practice  enables  one  to  distinguish  the  various 
simple  tones  which  enter  into  the  composition  of  a  clang,  such 
as  a  note  from  a  violin,  a  piano,  or  an  organ.  It  is  much  more 
difficult  to  distinguish  the  tones  which  enter  into  the  composi- 
tion of  a  vowel  sound  or  of  the  note  produced  by  a  singer.  The 
reason  may  be,  perhaps,  that  the  habit  of  perceiving  the  sound 
of  the  human  voice  as  a  whole  is  more  nearly  fixed.  The 
resonator  (Art.  801)  makes  it  easy  to  distinguish  the  compo- 
sition tones  of  almost  any  clang.  The  composition  tones  of 
a  clang  are  called  harmonic  overtones,  and  their  vibration  fre- 
quencies are  as  the  successive  whole  numbers  I,  2,  3,  etc.  This 
is  in  accordance  with  Fourier's  theorem,  and  is  fully  confirmed 
by  experiment.  The  various  overtones  in  a  clang  are  designated 
by  the  numbers  which  express  the  relation  of  their  vibration  fre- 
quency to  that  of  the  fundamental  tone.  Thus  we  speak  of  the 
fundamental,  the  first,  the  second,  third,  etc.,  tones  or  overtones 
of  a  clang.  That  quality  of  a  clang  which  depends  upon  the 
relative  intensities  of  the  various  overtones  is  called  timbre. 
The  tones  of  various  musical  instruments,  for  example,  owe 
their  peculiar  quality  largely  to  differences  in  timbre. 


CHAPTER   XIV. 
FREE   SONOROUS   VIBRATIONS. 

783.   Vibrations  of  air  columns.  —  Let  AB  (Fig.  539)  be  an 
indefinitely  long  tube  filled  with  air,  and  open  at  the  end  B. 


i             i 

A 

K—  rj- 

1    ya 

1 

!VB 

Fig. 

539. 

Consider  a  simple  train  of  sound  waves  of  wave  length  X  advan- 
cing in  the  tube  from  A  towards  B.  When  this  train  reaches 
B,  it  is  almost  entirely  reflected,  without  change  of  phase  (Art. 
623).  The  reflected  train  superposed  upon  the  advancing  train 
produces  a  stationary  train  (Art.  623)  of  which  the  nodes  are 

distant  -  from  each  other,  the  first  node  being  at  a  distance 
of  -  from  the  open  end,  which  is  an  antinode,  as  shown  by  the 

dotted  wave  lines. 

The  air  in  each  vibrating  segment  of  this  stationary  wave 
surges  back  and  forth  in  the  direction  of  the  tube ;  the  air  on 
the  two  sides  of  a  node  surges  towards  the  node  simultaneously, 
and  then  away  from  it  simultaneously,  so  that  the  air  at  the 
node  is  alternately  compressed  and  expanded,  but  does  not 
move.  The  time  r  required  for  one  back  and  forward  move- 
ment, that  is,  for  one  vibration  of  the  inclosed  air,  is  the  period 
of  the  wave  train.  From  equation  (320),  Art.  618,  we  have 

\  =  TV,  (i) 


FREE   SONOROUS   VIBRATIONS.  i$$ 

in  which  v  is  the  velocity  of  progression  of  sound  waves  in  the 
tube.  This  velocity  is  sensibly  the  same  as  the  velocity  in  the 
open  air.  The  frequency /of  the  vibration  of  the  inclosed  air 

is  equal  to  -;  whence,  using  the  value  of  r  from  equation  (i), 
we  have 

/-J  do 

Air  column  open  at  both  ends. — After  the  stationary  wave 
train  (or  vibration)  is  once  established  in  the  tube  A  £  (Fig.  539), 
the  tube  may  be  (ideally)  cut  across  at  any  antinode  without 
altering  *  the  subsequent  action  in  the  detached  portion  of  the 
tube  ;  except  that  the  vibrations  will  soon  die  away  as  the  energy 
of  the  vibrations  is  dissipated.  This  dissipation  is  partly  be- 
cause of  friction  against  the  walls  of  the  tube,  and  partly  because 
of  the  incomplete  reflections  from  the  open  ends  of  the  tube. 
The  tube  being  cut  across  at  an  antinode,  the  length  /  of  the 
detached  portion  will  be 

,     n\  -  ,... 

/  =  —,  (m) 

in  which  n  is  any  whole  number.  Therefore,  substituting  the 
value  of  X  from  (iii)  in  (ii),  we  have 

/=«J7  (343) 

in  which  f  is  the  frequency  of  vibration  of  an  air  column  of 
length  /  open  at  both  ends,  v  is  the  velocity  of  sound  in  air,  and 
n  is  any  whole  number. 

When  n  =  i,  the  frequency  of  vibration  of  the  air  column  is 
least,  and  the  tone  produced  is  called  the  fundamental  tone 
of  the  column.  When  n  is  2,  3,  etc.,  the  tone  produced  is 
called  the  second,  third,  etc.,  harmonic,  or  overtone.  The  char- 

*  The  wave  train  reflected  from  B  will  be  again  reflected  without  change  of  phase 
from  the  cut,  and  this  second  reflected  train  takes  the  place  of  the  original  advancing 
train. 


ELEMENTS   OF   PHYSICS. 

acter  of  the  vibration  when  «  =  i,  when  n  =  2,  and  when  n  —  3 
is  shown  by  the  dotted  wave  lines  in  Fig.  540. 

Air  column  closed  at  one  end.  —  When  the  stationary  wave 
train  is  once  established  in  the  tube  AB  (Fig.  539),  a  rigid  dia- 


Fig.  540. 

phragm  may  be  (ideally)  placed  across  the  tube  at  any  node 
without  altering  the  subsequent  behavior ;  except  that  the 
vibrations  will  soon  die  away.  In  this  case,  the  length  /  of 
the  detached  portion  of  the  tube  will  be 

.     n\  r  . 

/  =  -— ,  (iv) 

4 

in  which  n  is  any  odd  number.  Substituting  the  value  of  X  from 
(iv)  in  (ii),  we  have 

/=»T7  (344) 

4/ 

in  which  f  is  the  frequency  of  vibration  of  an  air  column  of 
length  /,  closed  at  one  end.  Since  n  in  this  case  is  necessarily 
an  odd  number,  an  air  column  open  at  one  end  has  only  odd 
harmonics.  The  character  of  the  vibrations  when  n  =  I,  when 
;/  =  3,  and  when  n  =  5  is  shown  by  the  dotted  wave  lines  in 
Fig.  541. 

784.  Simple  and  compound  vibrations  of  an  air  column.  —  An 
air  column  vibrating  so  as  to  give  its  fundamental  or  one  of  its 
harmonics  alone  is  said  to  vibrate  simply.  An  air  column  gen- 


FREE   SONOROUS   VIBRATIONS. 


157 


erally  vibrates  so  as  to  sound  its  fundamental  and  its  various 
harmonics  or  overtones  simultaneously,  thus  producing  a  clang. 
In  such  a  case  the  vibration  of  the  air  column  is  said  to  be  com- 


W7////A 


Fig.  541. 

pound.  The  relative  loudness  of  the  various  overtones  depends 
upon  the  shape  of  the  column  and  upon  the  manner  in  which 
the  vibrations  are  excited.  This  is  exemplified  in  the  strikingly 
different  clangs  of  organ  pipes,  whistles,  horns,  and  clarionets. 

785.  Organ  pipes.  —  The  vibrating  elements  of  the  pipe  organ 
are  columns  of  air  called  organ  pipes.  These  are  very  similar 
to  the  bark  whistles  known  to  children.  A  metal 
or  wooden  tube  AB  (Fig.  542)  incloses  the  vibrat- 
ing air  column,  and  the  vibrations  are  excited  by 
an  air  jet  which,  issuing  from  a  narrow  slit,  blows 
across  the  opening  against  the  sharp  edge  s.  This 
air  jet  makes  a  slight  noise,  which  starts  the  air 
column  vibrating  feebly.  These  vibrations  react 
upon  the  air  jet  and  cause  it  to  play  from  one  side 
to  the  other  of  s.  This  reinforces  the  vibrations, 
so  that  they  quickly  become  energetic.  When  the 
end  A  is  closed,  the  air  column  vibrates  in  accord- 
ance with  equation  (344),  and  only  the  odd  over- 
tones are  present.  When  the  end  A  is  open,  the  air  column 
vibrates  in  accordance  with  equation  (343),  and  both  even  and 


Fig.  54-2. 


158 


ELEMENTS   OF   PHYSICS. 


odd  overtones  are  present.  With  broad  pipes  the  overtones 
are  very  weak,  and  the  tone  approaches  a  pure  tone  in  char- 
acter. With  narrow  pipes  the  overtones  up  to  the  fifth  or  sixth 
are  audible.  When  an  organ  pipe  is  blown  strongly,  the  over- 
tones become  more  prominent.  If  blown  very  strongly,  the 
second  or  third  overtone  may  become  so  prominent  as  to  com- 
pletely dominate  the  tone. 

The  reed  pipe  is  an  air  column  into  which  a  stream  of  air 
flows  through  an  opening  in  and  out  of  which  a  spring  or  reed 
vibrates,  so  as  to  convert  the  air  stream  into  a  series  of  puffs 
of  the  proper  rhythm  to  excite  the  vibrations.  The  clarionet, 
the  cornet,  and  the  vocal  organs  of  man  are  types  of  the  reed 
pipe. 

786.  The  clarionet.  — The  tones  of  the  clarionet  are  produced 
by  the  vibrations  of  an  air  column,  the  length  of  which  may  be 
altered  at  will  by  uncovering  openings  in  the  side  of  the  tube. 
The  end  of  the  tube  which  is  placed  in  the  mouth  is  covered  by 
a  light  reed  or  tongue.  When  the  player  blows  into  the  instru- 
ment, this  reed,  acting  like  a  valve,  suddenly  closes  the  opening, 
and  a  wave  passes  down  the  tube,  is  reflected  from  the  open 
end  of  the  tube,  and,  returning,  strikes  the  reed  and  causes  it 
to  open  again.  The  impulse  is  then  repeated.  The  movement 
of  the  reed  is  not,  however,  entirely  controlled  by  the  surging 
of  the  air  in  the  tube  of  the  instrument.  The  lips  of  the  player 
are  pressed  against  the  reed  in  such  a  way  that  its  tendency 
is  to  open  and  close  in  the  proper  rhythm,  independently  of 
the  surging  of  the  air. 

The  abrupt  motion  of  the  reed  of  the  clarionet,  as  it  opens 
and  closes  the  mouth  end  of  the  instrument,  is  very  far  removed 
from  simple  vibration.  The  result  is,  that  the  various  (odd)  har- 
monics, as  well  as  the  fundamental  tone  of  the  air  column,  are 
sounded  quite  loudly,  and  the  instrument  gives  a  characteristic 
clang,  which  is  strikingly  different  from  the  sound  produced  by 
the  flute  or  the  organ  pipe. 


FREE    SONOROUS   VIBRATIONS. 


59 


787.  The  cornet.  —  In  the  case  of  the  cornet  the  lips  of  the 
performer  take  the  place  of  the  reed  of  the  clarionet,  and  the 
length   of   the  vibrating   column  of   air  is  altered  at  will   by 
means   of  valves   or   keys   which  include  or  exclude  auxiliary 
lengths  of  tube.     The  keys  provide  for  six  distinct  lengths  of 
air  column  and  by  sounding  the  various  harmonics  of  these  six 
lengths   at  will,  the  player   produces  any  desired  note.     The 
bugle  is  similar  to  the  cornet ;  except  that  it  has  a  tube  of  fixed 
length,  the  harmonics  of  which  are  sounded  at  the  will  of  the 
player.     The  harmonics  ordinarily  used  are  the  2d,  3d,  4th,  5th, 
and  6th. 

788.  The  vocal  organs.  —  These  consist  of  the  vocal  cords, 
two  muscular  membranes  which  are  stretched  across  the  top  of 
the  windpipe,  and  the  motith  cavity.     The  vocal  cords  are  tuned 
to  any  desired  pitch  by  muscular  effort  and  are  set  vibrating  by 
air  forced  from  the  lungs. 

The  smooth  tones  produced  in  singing  arise  from  vibrations 
in  which  the  vocal  cords  do  not  strike  against  each  other.  In 
speech  the  cords  strike  against  each  other  as  they  vibrate  and 
produce  sounds  which  contain  a  great  many  simple  component 
tones.  The  vowel  sounds  are  produced  by  so  shaping  the 
mouth  cavity  as  to  strengthen  (by  resonance)  certain  of  these 
component  tones  at  will.  The  consonant  sounds,  so  common 
in  speech,  are  characteristic  noises  with  which,  in  articulation, 
the  vowel  sounds  are  begun  and  ended. 

789.  Longitudinal  vibrations  of  rods  and  strings.  —  The  lon- 
gitudinal vibrations  of  a  rod  free  at  both  ends  are  similar  to 
those  of  an  air  column  open  at  both  ends.     The  frequency  of 
vibration  is  given  by  equation  (343),  in  which  v  is  the  velocity  of 
longitudinal  waves  (sound  waves)  along  the  rod.     The  dotted 
wave  lines  in  Fig.  540  show  the  character  of  the  longitudinal 
vibration  of  a  rod  with  one,  two,  and  three  nodes,  respectively. 
A  string  stretched  between  rigid   supports  vibrates   longitudi- 


ELEMENTS   OF   PHYSICS. 

rially  in  a  manner  similar  to  the  vibrations  of  an  air  column 
closed  at  both  ends. 

790.  Kundt's  experiment.  —  A  rod  (Fig.  543)  is  supported, 
say  at  its  center,  and  set  vibrating  longitudinally  by  rubbing  it 
with  a  rosined  cloth.  One  end  of  this  vibrating  rod  extends 
loosely  into  a  tube  of  air  AB.  A  train  of  waves  passes  out  from 
the  end  R  of  the  rod  as  it  vibrates  back  and  forth,  and  this  wave 
train  upon  reflection  from  the  closed  end  B  forms  a  stationary 


Fig.  543. 

train.  When  the  rod  is  adjusted  until  its  end  is  near  a  node, 
this  stationary  train  acquires  great  intensity.  Lycopodium  or 
other  light  powder,  which  is  strewn  inside  the  tube  AB,  is  swept 
out  of  the  vibrating  segments  by  the  surging  motion  of  the  air 
and  is  heaped  up  at  the  nodes.  The  distance  between  nodes 
is  then  easily  measured.  This  distance  is  half  the  distance 
traveled  by  the  sound  waves  in  the  air  of  the  tube  during  one 
vibration  of  the  rod.  If  the  rod  is  giving  its  fundamental  tone, 
for  which  n  =  I,  the  length  of  the  rod  is  half  the  distance 
traveled  by  a  sound  wave  in  the  rod  during  one  vibration. 
Therefore  the  ratio  of  the  length  of  the  rod,  divided  by  the 
distance  between  nodes  in  AB,  is  equal  to  the  velocity  of  sound 
in  the  material  of  the  rod  divided  by  its  velocity  in  air.  Know- 
ing the  velocity  of  sound  in  air,  its  velocity  in  the  rod  is  thus 
easily  determined.  The  tube  AB  may  then  be  filled  with  any 
gas  and  the  distance  between  the  nodes  again  measured; 
whence  the  velocity  of  sound  in  the  gas  may  be  found.  Some 
of  the  velocities  given  in  Art.  613  were  determined  in  this  way. 

791.  The  transverse  vibration  of  strings.  Preliminary  state- 
ment.—  If  a  stretched  string  be  struck  or  distorted  and  released, 
the  wave  produced  will  travel  at  a  velocity 


FREE    SONOROUS   VIBRATIONS.  !6i 

in  which  T  is  the  tension  of  the  string  in  dynes  and  m  is  its 
mass  per  unit  length. 

Let  AB  (Fig.  544)  be  an  indefinitely  long  stretched  string 
fixed  to  a  rigid  support  at  B.     Consider  a  simple  transverse 

A 


Fig.  544. 

wave  train  of  wave  length  \  advancing  from  A  towards  B. 
Upon  reflection  at  B  (with  change  of  phase)  a  stationary  wave 

train  will  be  formed,  of  which  the  nodes  are  distant  —  from  each 

2 

other.  Once  this  stationary  train  is  established,  a  rigid  support 
may  be  placed  under  the  string  at  any  node,  giving  a  vibrating 
string  of  which  the  length  is 

/      ^  /-\ 

/  =  —  ,  (11) 

in  which  n  is  any  whole  number.  The  time  r  of  one  complete 
vibration  of  the  string  is  equal  to  the  period  of  the  wave  train  ; 

so  that  \  =  TV,  and  the  frequency  of  the  vibrations  is  /=-• 
We  have,  therefore, 


Substituting  the  value  of  X  from  (ii),  and  the  value  of  v  from 
(i)  in  (iii),  we  have 


in  which  f  is  the  frequency  of  vibration  of  a  string  of  length  7, 
stretched  with  tension  T,  and  weighing  m  grams  per  centi- 
meter ;  and  n  is  any  whole  number. 

When  n  is  unity,  the  whole  string  is  one  vibrating  segment. 
The  tone  given  in  this  case  is  the  fundamental  of  the  string. 
When  n  equals  2,  3,  or  4,  etc.,  the  string  has  2,  3,  or  4,  etc., 
vibrating  segments.  The  tone  given  by  the  string  when  n  is 


UNIVERSITY 


1 62 


ELEMENTS   OF   PHYSICS. 


greater  than  unity  is  called,  as  in  the  case  of  vibrating  air  col- 
umns, a  harmonic  of  the  string,  or  an  overtone. 

792.  Simple  and  compound  vibrations  of  strings.  —  A  string 
vibrating  so  as  to  give  one  of  its  harmonics,  or  its  fundamental, 
is  said  to  vibrate  simply.  A  string  may  (and  generally  does) 
vibrate  so  as  to  give  simultaneously  its  fundamental  and  various 
harmonics.  In  such  a  case  its  vibration  is  said  to  be  compound. 
The  relative  intensities  of  the  fundamental  and  the  various  har- 
monics of  a  vibrating  string  depend  upon  the  manner  of  excit- 
ing the  vibrations.  Thus  the  plucked  string  of  the  guitar,  the 
struck  string  of  the  piano,  and  the  bowed  string  of  the  violin 
give  very  different  clangs,  and  these  clangs  change  very  percep- 
tibly with  the  point  at  which  the  string  is  plucked  or  struck 
or  bowed.  A  guitar  string  plucked  near  the  center  gives  only 


Fig.  545. 

odd  harmonics  and  those  not  strongly.  If  plucked  about  one- 
seventh  of  the  length  of  the  string  from  one  end,  the  overtones 
up  to  the  sixth  are  prominent,  and  the  clang  is  correspondingly 
rich. 

The    use   of   the   bow   enables   the    skilled   performer   upon 


FREE   SONOROUS   VIBRATIONS. 


163 


stringed  instruments  to  give  a  great  variety  of  complex  vibra- 
tions to  the  strings.  Figure  545  shows  a  few  curves  illustrating 
the  character  of  vibration  of  bowed  strings.  They  are  selected 
from  the  numerous  traces  published  by  Krigar-Menzel  and  Raps.* 
The  curves  were  obtained  by  photographing  upon  a  moving  plate 
the  motions  of  an  isolated  point  (or  element)  of  the  string. 

793.  The  use  of  sounding  boards.  —  A  vibrating  string  pro- 
duces a  comparatively  feeble  sound  when  it  is  stretched  over 
rigid  supports  upon  a  metal  bed,  because  of  the  very  small  dis- 
turbance of  the  air  produced  by  the  string  directly.     Stringed 
instruments  are  therefore  provided  with  thin  wooden  boards  or 
shells  to  which  the  bridges  which  support  the  strings  are  fixed. 
The  vibrations  are  transmitted  through  these  supports  to  this 
sounding  board,  and  thence  to  the  air. 

794.  Transverse  vibrations  of  stiff  rods   and   plates. — The 

velocity  of  propagation  of  a  transverse  wave  (a  bend)  along  a 
stiff  rod  or  plate  varies  with  the  wave  length.  The  frequencies 
of  the  various  simple  modes  of  vibra- 

-f-  very  weak. 

tion  of  a  rod  or  of  a  plate  are  not  so    ~Q 

—jf 9 strong  tremulo. 

simply  related,  as  are  those  of  vibrat-  :fi)     fo  weak. 

ing   air   columns   and    strings.     The  ^       -j-          very  strong. 

compound  vibration  of  a  stiff  rod  or  ±L 

of  an  elastic  plate  gives  therefore  a  -^- — ff-  ^e    weak 

clang   of    which    the   overtones    are 

.  .  Fig.  546. 

unharmonic.     The  discordant    sound 

produced  by  striking  a  steel  rod  and  the  sound  produced  by 
cymbals  and  gongs  are  examples.  The  overtones  of  the  tun- 
ing fork  and  of  the  bell  are  likewise  unharmonic.  Bell  found- 
ers have,  however,  learned  to  model  bells  in  such  a  way  as  to 
make  the  more  prominent  tones  harmonic.  A  bell  of  the  most 
approved  model  gives  a  rich  mellow  tone.  Figure  546  shows 
the  more  prominent  tones  of  a  Russian  bell  in  the  library  of 
Cornell  University. 

*  Wiedemann's  Annalen,  Vol.  44,  p.  623. 


164 


ELEMENTS   OF   PHYSICS. 


Chladni' s  figures.  —  The    nodal   lines    on   a   vibrating   plate 
may  be   shown  by  fixing   the   plate   horizontally   in   a  clamp, 

strewing  sand  upon  it,  and 
causing  it  to  vibrate  in  a 
simple  mode  by  means  of 
a  violin  bow.  The  sand 
collects  along  the  nodal 
lines.  These  sand  figures, 
which  were  discovered  by 
Chladni,  may  be  obtained 
in  a  great  variety  of  forms, 
corresponding  to  the  vari- 
ous simple  modes  of  vibra- 
tion of  the  plate.  To  this 
end  the  vibrations  of  the 
plate  must  be  controlled 
by  holding  the  fingers 
against  it  while  using  the 
bow.  Figure  547  shows 
some  of  the  figures  depicted  by  Chladni  in  his  treatise  on 
Acoustics.  (Leipzig,  1787.) 

795.  The  tuning  fork  is  a  stiff  rod  bent  into  the  form  shown 
in  Fig.  548.  The  character  of  its  fundamental  mode  of  vibration 
is  shown  by  the  lines  B.  The  two  nodes 
n,  n  are  near  together,  and  the  intervening 
segment,  together  with  the  metal  post  P, 
moves  up  and  down  through  a  small  am- 
plitude and  causes  the  sounding  board 
upon  which  the  fork  is  mounted  to  vibrate 
in  unison  with  the  fork.  The  second  tone 
of  a  fork,  which  is  some  three  or  four  oc- 
taves above  its  fundamental,  dies  out 
quickly  after  the  fork  is  thrown  into 
vibration,  leaving  the  fundamental  alone. 


Fig.  547. 


/    \ 


Fig.  548. 


FREE   SONOROUS   VIBRATIONS. 


I65 


The  tuning  fork,  therefore,  gives  a  pure  tone.  Carefully  tuned 
forks  are  much  used  by  musicians  as  standards  of  pitch.  (See 
Art.  781.) 

796.  Vibrating    diaphragms.  —  The    sound    produced    by    a 
vibrating  membrane  owes  its  peculiar  quality  largely  to  the 
characteristic  quickness  with  which  the  vibrations  die  out.     The 
sound  of  the  drum,  for  example,  is  so  brief  that  there  is  scarcely 
time  for  a  distinct  sensation  of  pitch  to  be  produced.     The  over- 
tones of  a  vibrating  membrane  or  diaphragm  are  unharmonic. 

A  diaphragm,  being  light  and  exposing  a  broad  surface  to  the 
air,  vibrates  very  perceptibly  with  the  air  when  any  sound 
strikes  it.  It  is  this  property  of  a  light  diaphragm  which  renders 
it  useful  in  the  telephone  and  the  phonograph. 

797.  Manometric  flames.  —  An  apparatus  has  been  devised 
by  Koenig  which  shows,  with  some  distinctness,  the  character 
of  the  vibrations  produced  by  a  sound.     It  depends  upon  the 
action  of  the  waves  upon  a  gas  flame.     A  thin  diaphragm  covers 


ACETYLENE 

Fig.  549. 


a  hole  in  the  side  of  a  chamber  through  which  gas  passes  to  a 
small  flame.  The  pressure  of  the  gas  fluctuates  with  the  move- 
ments of  the  diaphragm  and  causes  the  height  of  the  flame  to 
vary  accordingly.  When  the  flame  is  viewed  in  a  rotating  mir- 


!66  ELEMENTS   OF   PHYSICS. 

ror,  or  is  photographed  upon  a  moving  plate,  it  presents  a  saw- 
toothed  appearance  which  varies  with  the  character  of  the  sound 
falling  upon  the  diaphragm.  Photographs  of  the  manometric 
flame  are  shown  in  the  accompanying  plate.  These  photo- 
graphs were  taken  from  a  brilliant  acetylene  flame  burning  in 
oxygen.  The  movement  of  the  photographic  plate  was  such 
as  to  make  it  necessary  to  read -the  figure  from  right  to  left. 
The  apparatus*  used  in  taking  these  photographs,  which  is 
shown  in  Fig.  549,  is  that  devised  by  Merritt  in  1893.  The 
lens  forms  an  image  of  the  flame  upon  the  sensitive  plate 
AB,  which  moves  rapidly  in  a  direction  perpendicular  to  the 
paper. 

*  Physical  Review,  Vol.  I.,  p.  166. 


CHAPTER   XV. 
IMPRESSED   VIBRATIONS   AND   RESONANCE. 

798.  Proper  vibrations  ;    impressed  vibrations.  —  The  vibra- 
tions which  a  body  performs  when  struck  or  disturbed  in  any 
way,  and  left  to  itself,  are  called  its  proper  vibrations. 

When  a  simple  train  of  sound  waves  of  any  wave  length 
strikes  a  body,  the  body  is  made  to  vibrate  in  unison,  or  in  the 
same  rhythm,  with  the  impinging  waves.  Such  vibrations  are 
called  impressed  vibrations.  A  compound  wave  train  causes  a 
body  to  perform  simultaneously  the  simple  vibrations  corre- 
sponding to  the  various  simple  wave  trains  which  enter  into  the 
composition  of  the  compound  train. 

The  quickness  with  which  a  body  assumes  a  steady  state  of 
impressed  vibration  under  the  action  of  sound  waves  depends 
upon  the  mass  of  the  body  and  upon  the  extent  of  the  surface 
which  it  exposes  to  the  action  of  the  waves. 

The  violence  of  the  impressed  vibrations  depends  upon  the 
intensity  of  the  impinging  waves;  upon  the  extent  to  which  the 
vibrations  are  damped;  and  upon  the  relation  between  the  fre- 
quency of  the  impinging  waves  and  the  frequency  of  the  proper 
vibrations  of  the  body. 

799.  Damping. — Vibrations  are  said  to  be  damped  when  they 
die  out  quickly.     This  is  generally  due,  in  part,  to  the  dissipa- 
tion of  energy  in  the  body  in  the  form  of  heat,  as  it  is  repeat- 
edly distorted,  and  in  part  to  the  giving  up  of  energy  to  the 
surrounding  air.     Thus  the  vibrations  of  light   bodies,  which 

expose  considerable  surface  to  the  air  (diaphragms,  etc.),  and 

167 


T68  ELEMENTS    OF   PHYSICS. 

of  bodies  which  are  imperfectly  elastic,  are  greatly  damped. 
On  the  other  hand,  the  vibrations  of  a  heavy,  elastic  body, 
such  as  a  tuning  fork,  are  but  slightly  damped.  A  heavy 
tuning  fork  performs  several  thousands  of  perceptible  vibrations 
when  struck.  The  column  of  air  in  an  organ  pipe  performs 
several  hundreds  of  perceptible  vibrations  after  the  exciting 
cause  ceases.  A  drum  head  performs  only  a  very  few  per- 
ceptible vibrations  when  struck. 

800.  Resonance.  —  Very  perceptible  vibrations  are  impressed 
upon  diaphragms  and  stretched  membranes  by  sound  wave 
trains  of  any  frequency.  The  vibrations  impressed  lipon  a 
heavy  body,  of  which  the  damping  is  slight,  are  much  more 
violent  when  the  impressed  frequency  approaches  the  proper  fre- 
quency of  the  body.  Thus  the  sound  of  a  tuning  fork  (removed 
from  its  sounding  board)  is  perceptibly  enforced  when  it  is  held 
near  the  open  end  of  a  tube  of  any  length.  If  the  length  of  the 
air  column  is  adjusted,  for  example,  by  pouring  water  into  the 
tube,  the  sound  becomes  louder  as  the  proper  frequency  of 
the  air  column  approaches  that  of  the  fork;  and  it  reaches  a 
very  distinct  maximum  when  the  impressed  vibrations  become 
proper  to  the  air  column.  With  bodies  which  exhibit  less  and 
less  damping  the  maximum  violence  of  impressed  vibrations  at 
proper  frequency  becomes  more  and  more  pronounced;  and  at 
the  same  time  the  impressed  vibrations  of  improper  frequency 
become  more  and  more  nearly  imperceptible.  Thus,  a  massive 
tuning  fork,  mounted  upon  its  sounding  board,  is  thrown  into 
quite  violent  vibration  by  a  tone  of  its  proper  frequency  sus- 
tained for  four  or  five  seconds. 

This  pronounced  maximum  violence  of  impressed  vibration 
at  proper  frequency  is  called  resonance,  and  the  vibrating  body 
or  air  column  is  called  a  resonator. 

When  impressed  vibrations  are  proper  to  a  body,  the  action  of 
the  impulse  due  to  each  successive  wave  is  to  add  to  the  exist- 
ing motion,  and  the  vibrations  increase  in  violence  until  the 


IMPRESSED   VIBRATIONS   AND   RESONANCE.  iftg 

energy  given  to  the  vibrating  body  by  the  impinging  waves 
is  all  dissipated  by  damping.  It  is  for  this  reason  that  the 
impressed  vibrations  become  quite  violent  when  the  damping 
is  small.  On  the  other  hand,  when  improper  vibrations  are 
impressed  upon  a  body,  the  periodic  forces,  with  which  the 
waves  act  upon  the  body,  must  take  the  place  more  or  less  of 
the  elastic  forces,  which  ordinarily  cause  a  body  to  vibrate. 

801.  Analysis  of  clangs  by  means  of  resonators.  —  Any  over- 
tone of  a  clang  may  be  easily  detected  or  brought  to  notice  by 
intermittently  strengthening  it  by  means  of  a  resonator  tuned 
to  unison  with  it. 

A  convenient  resonator  for  this  purpose  is  made  by  drawing 
the  end  of  a  glass  tube  to  such  size  as  will  fit  tightly  into  the 
ear.  The  other  end  of  the  tube  is  left  open.  The  inclosed  air 
will  strengthen  very  perceptibly  any  tone  in  unison  with  it. 
The  action  is  more  striking  if  the  tube  is  repeatedly  removed 
from  the  ear  and  replaced.  For  tones  of  low  pitch  such  a 
tubular  resonator  would  be  unwieldy.  In  this  case  it  is  more 
convenient  to  use  a  hollow  globular  vessel  of  glass  or  metal,  with 
a  small  neck  on  one  side  to  project  into  the  ear  and  an  opening 
on  the  other  side  several  centimeters  in  diameter.  In  order  to 
analyze  a  clang,  one  after  another  of  a 
series  of  such  resonators  is  applied  to 
the  ear.  Figure  550  shows  an  ad- 
justable resonator  designed  by  Koenig. 
It  consists  of  two  cylindrical  brass  F.  550 

tubes  fitted-  to  one  another.    The  outer 

tube  is  contracted  to  a  narrow  opening  which  enters  the  outer 
ear  of  the  observer.  The  other  tube  is  partially  closed  by  a  cap, 
which  contains  a  circular  opening  (1-3  centimeters  in  diameter). 

802.  Vowel  sounds.  —  The  various  vowel  sounds  are  charac- 
terized each  by  one  or  two  tones  of  definite  pitch.     In  producing 
a  given  vowel,  the  mouth  cavity  is  shaped  so  as  to  strengthen  by 


I/O 


ELEMENTS   OF   PHYSICS. 


resonance  the  tones  which  characterize  the  vowel.  The  charac- 
teristic tones  of  some  of  the  vowels,  as  determined  by  Helm- 
holtz,  are  as  follows  : 


VOWEL. 

TONE. 

VIBRATION  FREQUENCY.* 

u  as  in  rude 

/ 

173 

6  as  in  no 

c" 

517 

a  as  in  paw 

g" 

775 

a  as  in  part 

d\>'" 

1096 

a  as  y&pay 

f  and  tip" 

346  and  1843 

e  as  in  pet 

c"" 

2068 

e  as  in  see 

f  and  d"" 

173  and  2322 

Figure  551  gives  these  characteristic  tones  of  the  vowels  in 
terms  of  the  ordinary  musical  notation. 

In  ordinary  speech  the  rough  sound  from  the  vocal  cords 
easily  excites  the  proper  resonance  in  the  mouth  cavity  for  the 


/              » 

uw 

li 

1 
u        6        a         a 

f 

a         e        e 

Fig.  551. 

production  of  any  required  vowel.  The  smooth  tone  of  a  singer, 
however,  may  not  contain  the  characteristic  tones  of  a  vowel,  so 
that  these  cannot  be  strengthened  by  resonance.  In  this  case 
those  overtones,  of  the  note  which  is  sung,  which  are  nearest 
the  characteristic  tones  of  a  vowel,  for  which  the  mouth  cavity 
is  set,  are  strengthened,  and  in  this  way  the  vowel  sound  is 
(incompletely)  characterized.  It  is  a  well-known  fact  that 
spoken  words  are  much  more  easily  understood  than  words 

*  Complete  vibrations. 


IMPRESSED   VIBRATIONS   AND   RESONANCE. 


I/I 


which  are  sung.  The  difference  in  distinctness  is  due  largely 
to  the  imperfect  character  of  vowels  when  sung.  Overtones  of 
a  given  pitch  are  more  widely  separated  in  a  note  of  high  pitch 
than  in  a  note  of  low  pitch,  so  that  the  mouth  cavity,  shaped  to 
give  the  characteristic  tone  of  a  vowel,  is  less  likely  to  produce 
the  desired  effect  with  high  notes  than  with  low.  The  words 
of  a  soprano  singer  are,  in  fact,  less  distinct  than  the  words  of  a 
bass  singer  of  similar  schooling. 

The  proper  tones  of  the  mouth  cavity  for  the  production  of 
the  vowels  o,  a,  and  a  may  be  easily  heard  by  thumping  against 
the  cheek  when  the  mouth  is  prepared  to  sound  those  vowels. 
Helmholtz  determined  the  characteristic  tones  of  the  vowels  by 
finding  which  of  a  series  of  tuning  forks  placed  before  the 
mouth  in  succession  would  produce  resonance  in  the  mouth 
cavity  shaped  to  produce  a  given  vowel.  He  also  determined 
by  the  help  of  resonators,  as  described  in  the  previous  article, 
the  actual  tones  present  in  the  various  vowel  sounds ;  and  by 
combining  the  sounds  of  suitable  tuning  forks  he  was  able  to 
imitate  successfully  these  various  vowels. 

803.  The  artificial  reproduction  of  speech.  The  phonograph. 
—  Vowel  sounds,  and  indeed  all  the  sounds  used  in  speech,  can 
be  reproduced  by  means  of  any  device  which  is  capable  of  giving 
out  the  necessary  combination  of  simple  tones  with  the  proper 
relative  intensities.  Such  an  instrument  is  the  phonograph  (and 
its  modification  the  gramophone). 

It  consists  essentially  of  a  thin  diaphragm,  to  which  is  fastened 
a  light  tool  which  scratches  a  minute  groove  in  a  rotating 
smooth  cylinder  made  of  a  hard  wax-like  compound,  or  of  soap. 
A  sound  striking  the  diaphragm  impresses  vibrations  upon  it, 
and  causes  the  attached  tool  to  cut  a  groove  of  varying  depth. 
A  record  of  the  sound  is  thus  made  upon  the  cylinder.  The 
cylinder  is  driven  with  a  very  nearly  uniform  motion  of  rotation 
by  means  of  an  electric  motor.  In  some  cases  clockwork  is 
employed. 


1/2 


ELEMENTS   OF   PHYSICS. 


To  reproduce  the  sound  a  round-ended  tool,  which  is  attached 
to  the  diaphragm,  is  adjusted  to  follow  this  groove  and  the  cyl- 
inder is  set  rotating  at  its  former  speed.  The  varying  depth  of 
the  groove  causes  the  diaphragm  to  vibrate  and  the  sound  is 
reproduced.  The  phonograph  may  be  regarded  as  a  develop- 
ment of  an  earlier  instrument,  the phonautograph,  a  mechanism  by 
means  of  which  a  curve,  showing  the  character  of  the  vibrations 
producing  a  sound  or  produced  by  the  sound,  is  traced.  The 
movements  of  a  diaphragm  are  transmitted  to  a  tracing  point 
which  marks  a  line  upon  smoked  paper.  The  paper  is  carried 
on  a  rotating  cylinder. 


CHAPTER   XVI. 
THE   EAR   AND    HEARING. 

804.  The  human  ear.*  —  The  fact  that  the  various  overtones 
of  a  clang  may  be  distinctly  heard,  shows  that  the  conception 
of  simple  and  compound  vibrations  as  developed  in  Arts.  773  and 
774  is  not  merely  a  mathematical  fiction,  but  that  it  has  real 
physical  significance.  This  significance  is  briefly  this  ;  namely, 
that  a  body,  of  which  the  proper  vibration  frequency  is  in 
unison  with  any  simple  tone  of  a  clang,  is  set  into  violent 
vibration  by  the  clang.  This  property  has  been  discussed  in 
the  articles  on  Resonance  (Chapter  XV.). 

It  was  first  pointed  out  by  Helmholtz  that  our  perception  of  the 
various  simple  tones  in  a  clang  must  depend  upon  the  existence 
of  a  series  of  organs  (the  end  organs  of  the  auditory  nerves)  in 
the  ear,  each  of  which  has  a  proper  vibration  frequency  and  is 
sensitive  (by  resonance)  to  simple  tones  nearly  in  unison  with 
it.  This  action  may  be  illustrated  by  means  of  the  piano,  as 
follows : 

A  musical  sound  of  characteristic  timbre,  for  example,  a 
vowel  sound,  is  sung  loudly  against  the  sounding  board  of 
a  piano,  of  which  the  damper  is  raised  so  as  to  leave  the 
strings  free.  Those  strings  which  are  in  unison  with  the 
various  simple  tones  of  the  clang  are  set  into  vibration  and  we 
hear  a  continuation,  by  the  piano,  of  the  vowel  sound  after  the 
singing  ceases.  Imagine  each  string  of  a  piano  to  be  connected 

*  The  anatomy  of  the  ear  is  too  complex  to  be  described  here  with  any  fullness. 
See  Helmholtz,  Die  Lehre  von  den  Tonempfindungen,  pp.  209-250. 

173 


174 


ELEMENTS   OF   PHYSICS. 


to  a  nerve  fiber,  and  we  have  an  apparatus  which  would  perceive 
sounds  as  they  are  actually  perceived  by  the  ear. 

The  process  of  perception  is  as  follows :  Sound  waves  enter 
the  ear  and  strike  against  the  tympanic  membrane.  The  vibra- 
tions of  this  membrane,  of  which  the  area  is  about  70  square 
millimeters,  are  reduced  in  amplitude  and  concentrated*  upon 
another  diaphragm,  of  about  5  square  millimeters  in  area,  by 
the  action  of  a  chain  of  three  tiny  bones.  This  second  dia- 
phragm covers  a  small  window  (the  oval  window]  of  a  bony 
cavity,  called  the  labyrinth,  which  is  filled  with  a  watery  fluid. 
Another  opening  of  the  labyrinth,  the  round  window,  is  cov- 
ered with  a  diaphragm  which  is  entirely  free.  The  vibrations 
of  the  diaphragm  in  the  oval  window  cause  the  watery  fluid  of 
the  labyrinth  to  surge  back  and  forth  between  the  two  windows 
and  through  the  chambers  of  the  labyrinth.  One  of  these  cham- 
bers, the  cochlea,  is  very  long,  and  is  coiled  upon  itself  like  a 
snail  shell.  The  end  organs  of  the  auditory  nerves  are  located 
in  membraneous  structures,  which  are  in  part  suspended  in  the 
watery  fluid  of  the  labyrinth,  and  in  part  constitute  an  elastic 
diaphragm,  the  basilar  membrane,  which  divides  the  cavity  of 
the  cochlea  longitudinally.  The  various  shreds  of  this  basilar 
membrane  seem  to  be  the  resonating  elements  of  the  ear. 

Remark.  — Those  shreds  of  the  basilar  membrane  which  are 
in  unison  with  a  given  simple  tone  are  most  strongly  excited 
by  the  tone ;  and  the  intensity  with  which  an  adjacent  shred  is 
excited  falls  off  as  its  proper  vibration  frequency  differs  more 
and  more  from  the  vibration  frequency  of  the  tone.  The  pro- 
duction of  audible  beats  by  the  interference  of  two  tones 
depends  upon  the  simultaneous  action  of  the  two  tones  upon 
the  same  shreds  of  the  basilar  membrane.  When  the  two  tones 
work  together  to  produce  commotion  upon  the  shreds  which 
they  both  affect,  the  sound  is  a  maximum,  and  when  they  are 
opposite  in  phase  they  produce  but  little  commotion,  and  the 
sound  is  a  minimum. 

*  See  Helmholtz,  Loc.  cit. 


THE   EAR  AND   HEARING. 


175 


805.  Persistence  of  the  sensation  of  sound.  —  When  the  stimu- 
lation of  a  nerve  ceases,  the  accompanying  sensation  continues 
for  a  length  of  time,  which  depends  upon  the  intensity  of  the 
stimulation,    and  which   varies   greatly  with   different   nerves. 
Sensations    of   light   persist   much   longer   than  sensations  of 
sound.     A  periodic   light   becomes    sensibly  continuous  when 
the  flashes  follow  one  another  at  the  rate  of  forty  per  second. 
A  periodic  sound  —  for  example,  the  sound  of  a  tuning  fork  of 
high  pitch  which  is  shut  off  from  the  ear  intermittently  —  has 
been  found  by  Mayer   to   become   smooth  when   the   fluctua- 
tions reach  a  frequency  of   one   hundred   and   thirty-five   per 
second.     The  sensation  produced  by  an  intermittent  tone  is 
very  rough  and  unpleasant,  —  an  effect  which  is  called  discord, 
or  dissonance.      The  discordant  effect  increases  with  the  fre- 
quency of  fluctuation,  reaches   a  maximum,  and  finally  disap- 
pears, when  the  sensation  becomes  smooth. 

806.  Interference.  —  The   shortness   of   the   interval   during 
which  a  tone  sensation  continues  after  the  stimulation  ceases 
is  very  intimately  connected  with    the   phenomenon   of   inter- 
ference of  two  tones.     The  general  character  of  this  phenome- 
non is  described  in  Chapter  VIII. 

Consider  two  simple  tones  of  vibration  frequencies  /  and  /', 
which  are  very  nearly  equal.  At  a  certain  instant  the  wave 
trains  which  constitute  these  tones  will  be  in  like  phase  as  they 
enter  the  ear.  The  disturbance  produced  and  the  correspond- 
ing sensation  will  then  be  a  maximum. 

When  the  tone  of  higher  pitch  has  gained  half  a  vibration 
(or  half  a  wave  length)  over  the  other,  the  wave  trains  will  be 
opposite  in  phase  as  they  enter  the  ear,  and  the  sensation  will 
be  a  minimum.  When  the  higher  tone  has  gained  a  whole 
vibration,  the  waves  will  again  enter  the  ear  in  like  phase,  and 
the  sensation  will  again  be  a  maximum.  These  successive 
maximum  sensations  are  called  beats.  The  number  of  beats 
which  occur  in  one  second  is  f—f. 


ELEMENTS   OF   PHYSICS. 


As  the  pitch  of  one  of  the  tones  increases,  the  difference 
f—f  increases;  the  beats  become  more  frequent;  the  inter- 
mittent sound  sensation  becomes  more  disagreeable  or  dis- 
cordant, and  soon  reaches  a  point  of  maximum  discord,  after 
which  the  discord  decreases  again.  When  the  beats  become 
sufficiently  frequent,  the  sensation  becomes  smooth  because  of 
the  persistence  of  the  sound  sensations.  The  number  of  beats 
per  second  which  produces  maximum  discord,  and  the  number 
per  second  which  leaves  the  resulting  sensation  smooth,  vary 
with  the  absolute  pitch  of  the  interfering  tones  in  a  manner 
shown  approximately  in  the  following  table. 

FREQUENCY   OF   FLUCTUATIONS. 


VIBRATION  FREQUENCY. 

WHEN  TONE  BECOMES 
SMOOTH. 

WHEN  DISCORD  is  A 

MAXIMUM. 

64 

16 

6.4 

128 

26 

10-4 

256 

47 

18.8 

384 

60 

24.0 

5I2 

78 

31.2 

640 

90 

36.0 

768 

I09 

43-6 

1024 

135 

54.0 

807.  Combination  tones.  —  The  principle  of  superposition 
(Art.  621),  viz.,  that  two  tones  may  exist  simultaneously  with- 
out affecting  each  other,  is  true  only  when  the  forces  producing 
a  distortion  are  strictly  proportional  to  the  distortion.  In  this 
case  an  added  distortion  produces  exactly  the  same  additional 
forces  no  matter  what  the  initial  distortion  may  be.  No  actual 
material  satisfies  this  condition  except  for  very  small  distortions, 
a  fact  which  is  ordinarily  expressed  by  saying  that  all  materials 
are  imperfectly  elastic.  Two  weak  (primary)  tones  do  not  sen- 
sibly affect  each  other ;  but  as  they  grow  louder  certain  other 
tones,  produced  by  their  mutual  action,  are  found  to  accompany 


THE   EAR   AND    HEARING. 


177 


them.  These  accompanying  tones  are  called  combination  tones. 
Combination  tones  are  most  pronounced  in  those  cases  in  which 
the  primary  tones  cause  the  same  substance  or  the  same  portion  of 
air  to  vibrate  violently.  The  imperfectly  elastic  chain  of  bones 
in  the  ear,  for  example,  is  conducive  to  the  formation  of  com- 
bination tones. 

Difference  tones.  —  The  most  prominent  combination  tone  is 
that  of  which  the  frequency  is  equal  to  the  difference  of  the 
frequencies  of  the  two  primary  tones.  This  is  called  the  differ- 
ence tone  of  the  first  order.  This  difference  tone  forms  differ- 
ence tones  with  each  of  the  primary  tones,  in  like  manner. 
These  are  called  difference  tones  of  the  second  order;  and  so  on. 

Summation  tones. — Less  prominent  than  the  difference  tones 
is  the  combination  tone  of  which  the  frequency  is  equal  to  the 
sum  of  the  frequencies  of  the  two  primary  tones.  This  is  called 
a  summation  tone  of  the  first  order.  Summation  tones  of  the 
first  order  form  summation  tones  with  each  primary  tone. 
These  are  called  summation  tones  of  the  second  order. 

Difference  tones  of  the  first  order  are  very  noticeable  when 
two  tuning  forks  are  sounded  simultaneously.  Helmholtz  has 
shown  that  difference  tones  of  higher  order  and  summation 
tones  are  audible  in  the  sound  of  a  two-voiced  siren,  and  in 
the  sound  produced  by  two  notes  of  a  reed  organ  sounding 
simultaneously.  It  often  requires  the  help  of  a  resonator, 
however,  to  bring  them  into  notice. 

808.  Miscellaneous  phenomena  depending  upon  the  reflection, 
refraction,  and  diffraction  of  sound.  —  The  reflection,  refraction, 
and   diffraction    of   sound  waves   have   been   discussed  in  the 
earlier  chapters  of  this  volume.     Certain   phenomena  relating 
to  hearing  which  result  from  those  processes,  however,  are  still 
to  be  considered.     These  are  briefly  described  in  the  following 
articles. 

809.  Echo.  —  This  well-known  phenomenon  is  produced   by 
the  reflection  of   sound.     The  echo  from  the  side  of   a  large 


I78 


ELEMENTS   OF   PHYSICS. 


building  is  very  distinct,  and  the  smooth  face  of  a  cliff,  or  a 
well-defined  forest  front,  may  produce  an  echo  sufficiently  dis- 
tinct to  repeat  words.  If  the  reflecting  surface  is  sufficiently 
distant,  an  entire  sentence  may  be  repeated  by  the  echo. 

An  echo  grows  less  distinct  the  more  irregular  the  reflecting 
surface,  and  it  becomes  an  indistinct  roar  when  the  reflecting 
surface  is  very  irregular.  With  multiple  reflections,  as  in  the 
case  of  the  two  walls  of  a  tunnel  or  of  a  canon,  a  sharp  loud 
sound,  such  as  the  report  of  a  gun,  is  prolonged  in  a  manner 
resembling  thunder. 

Reflection  often  produces  the  effect  of  an  apparent  change  of 
direction  of  a  sound,  when  from  any  cause  the  direct  waves  are 
masked  or  diverted  from  any  cause,  so  that  the  hearer  perceives 
only  the  reflected  wave  trains. 

810.  The  influence  of  the  refraction  of  sound  upon  hearing  at  a 
distance. —  Phenomena  due  to  regular  refraction,  as  sound  waves 
pass  from  one  medium  to  another,  in  which  the  velocity  is  dif- 
ferent, do  not  occur  to  ordinary  observation.  The  following 
phenomena,  however,  which  are  due  essentially  to  refraction, 
are  of  common  occurrence. 

The  velocity  of  the  wind  is  usually  less  near  the  ground  than 
higher  up,  and  the  upper  portion  of  a  sound  wave  W  (Fig.  552), 


WIND 


GROUND 
Fig.  552. 


proceeding  against  the  wind,  is  retarded.  The  direction  of  pro- 
gression of  the  wave  is  thus  thrown  upwards  and  the  sound 
tends  to  leave  the  region  near  the  ground.  When  the  wave 


THE   EAR   AND   HEARING. 


179 


travels  with  the  wind,  the  tendency  is  to  concentrate  the  sound 
near  the  ground.  It  is  a  familiar  fact  that  it  is  much  more 
difficult  to  make  one's  self  heard  against  than  with  the  wind. 

Sound  travels  faster  in  hot  than  in  cold  air.  When  the  air 
near  the  ground  is  warmer  than  it  is  higher  up,  the  upper  por- 
tion of  a  sound  wave  is  retarded  and  the  sound  tends  to  leave 
the  ground.  When  the  air  near  the  ground  is  relatively  cool, 
the  tendency  is  for  the  sound  to  be  concentrated  near  the 
ground.  The  greater  distinctness  of  distant  sounds  by  night 
than  by  day  is  no  doubt  due  largely  to  this  cause. 

811.  The  influence  of  diffraction  upon  the  sense  of  direction  of 
a  sound.  —  Our  sense  of  the  direction  of  a  sound  seems  to  depend, 
in  part,  upon  diffraction.     We  have  this  sense  only  with  com- 
plex sounds,  not  with  a  sound  which  consists  of  a  single  simple 
wave  train.     The  approaching  complex  waves   reach   one   ear 
without  much  obstruction,  while  the  other  ear  is  more  or  less 
shaded  from  them  by  the  head.     This  shading  action  is  greater 
the  shorter  the  wave  length,  so  that   different  sensations  are 
produced  in  the  two  ears.     Differences  of   sensation    brought 
about  in  this  manner  are  significant  of  direction,  and  have  come 
to  be  perceived  as  such  without  entering  consciousness  in  any 
other  way. 

812.  Changes  of  pitch  due  to  the  motion  of  the  sounding  body 
or  of  the  hearer.  —  Since  pitch  depends  upon  the  frequency 
with  which  successive  waves  fall  upon  the  ear,  it  is  obvious  that 
motion  in  the  direction  of  the  wave  (either  with  or  against  the 
wave)  will  respectively  lower  or  raise  the  pitch.     The  velocity 
of  sound  is  so  small  that  the  motion  of  a  railway  train  is  by  no 
means  inappreciable  in  comparison.     Thus  the  whistle  of  an 
approaching  engine  is  distinctly  raised  and  that  of  a  receding 
train  is  distinctly  lowered  by  the  motion  of  the  sounding  body. 
The  effect  is  especially  noticeable  to  an  observer  upon  one  train 
who  listens  to  the  whistle  of  an  engine  passing  in  an  opposite 
direction. 


OF  THB 

'CTNIVERSITY 


!8o  ELEMENTS   OF   PHYSICS. 

813.   Sounds  of  all  wave  lengths  have  the  same  velocity. — 

We  have  very  direct  evidence  of  this  fact  in  the  case  of  both 
speech  and  music.  If  different  components  of  a  spoken  sound 
or  of  a  concerted  piece  traveled  at  different  velocities,  we  should 
have  sensations  varying  'with  the  distance  from  the  source. 
Both  speech  and  music  indeed  would  become  unrecognizable  at 
considerable  distances  because  the  various  components  to  which 
the  timbre  is  due  would  fail  to  reach  the  ear  simultaneously.* 
In  point  of  fact,  the  only  effect  of  distance,  aside  from  general 
loss  of  intensity,  is  to  diminish  the  loudness  of  sounds  varying 
in  pitch  by  somewhat  unequal  amount.  Some  waves  seem  to 
carry  further  than  others. 

*  This  remark,  although  true  of  light  and  sound,  fails  with  water  waves.     Let  A 
(Fig.  553)  represent  a  water  wave  produced  by  a  quick  movement  of  an  oar.     When 


Fig-  553. 

this  wave  has  traveled  a  short  distance,  it  will  be  seen  to  have  assumed  the  form 
shown  by  the  wave  line  B.  The  various  simple  wave  components  of  A  are  separated 
because  of  their  different  velocities.  The  movement  of  a  chip  produced  by  the  wave 
B  would  be  very  different  in  character  from  any  motion  which  could  possibly  pro- 
duce the  wave  A.  This  phenomenon  is  shown  very  beautifully  by  the  ripple  produced 
by  dipping  an  oar  edgewise  into  still  water. 


CHAPTER   XVII. 
MUSICAL    INTERVALS   AND   SCALES. 

814.  Pitch  intervals.  —  It  is  shown  in  the  next  following  arti- 
cle [815]  that  the  consonant  relation  of  two  tones  depends 
mainly  upon  the  ratio  of  their  vibration  frequencies.  Two 
pitch  intervals  are  therefore  said  to  be  equal  when  they  are 
expressed  by  the  same  frequency  ratio. 

Consider  a  number  of  tones  of  which  the  vibration  frequencies 
are  n,  an,  a2n,  a8n,  etc.  The  frequency  ratio  of  two  successive 
tones  in  this  series  is  a.  Let  their  pitch  interval  be  /.  The 
frequency  ratio  of  the  first  and  third  tones  is  #2,  and  their  pitch 
interval  will  be  2/ ;  the  frequency  ratio  of  the  first  and  fourth 
tones  is  a?,  and  their  pitch  interval  will  be  3/,  etc.  Let  /  be 
the  logarithm  of  a.  Then  2  /  is  the  logarithm  of  a2,  3  /  is  the 
logarithm  of  a3,  etc.  Therefore  we  have  for  these  various  pitch 
intervals  the  following  relation  : 

Frequency  ratios  a     a?      as      a*      etc. 

Logarithms  of  frequency  ratios      /     2  /     3  /     4  /     etc. 

.  Pitch  intervals  p     2p     *$p    4p    etc. 

It  will  be  seen  that  the  pitch  interval  between  two  tones  is  pro- 
portional to  the  logarithm  of  their  frequency  ratio. 

Pitch  intervals  are  ordinarily  expressed  in  terms  of  frequency 
ratios.  When  the  relative  magnitude  of  a  number  of  pitch 
intervals  is  the  object  of  consideration,  it  is,  however,  con- 
venient to  express  the  intervals  in  terms  of  the  logarithms  of 
their  frequency  ratios.  As  an  example,  see  the  discussion  of 
the  tempered  musical  scale.  (Art.  824.) 

181 


l$2  ELEMENTS   OF   PHYSICS. 

815.  Complete  and  approximate  consonance  of  compound  tones.* 
Preliminary  statement.  —  Tones  ordinarily  used  in  music  are 
compound.  The  fundamental  tone  usually  predominates,  in  a 
compound  tone,  and  the  overtones  2,  3,  4,  5,  6,  and  8  usually 
occur,  decreasing  in  loudness  in  the  order  given.  The  follow- 
ing discussion  of  consonance  is  limited  to  the  influence  of  these 
six  overtones. 

When  two  compound  tones  A  and  B  are  in  unison,  their 
respective  overtones  are  in  unison  also ;  the  combined  sound  of 
the  two  tones  is  entirely  free  from  roughness  due  to  beats,  and 
the  two  tones  are  completely  consonant.  When  the  two  tones 
are  not  in  unison,  then,  even  if  their  difference  in  pitch  is  so 
great  that  the  fundamental  tones  of  A  and  B  do  not  produce 
audible  beats,  some  of  the  overtones  of  A  will  generally  be 
near  enough  to  some  of  the  overtones  of  B  to  produce  marked 
roughness  in  the  combined  sound  of  A  and  B\  that  is,  to  pro- 
duce distinct  dissonance.  If  the  pitch  of  the  tone  B  is  slowly 
raised  or  lowered,  starting  from  unison  with  A,  this  dissonance 
passes  through  a  very  marked  minimum  value  every  time  one 
(or  more)  of  the  overtones  of  A  come  into  unison  with  one 
(or  more)  of  the  overtones  of  B.  The  two  tones  A  and  B  are 
approximately  consonant  when  their  dissonance  thus  reaches 
a  minimum  value. 

The  following  example  will  make  this  clear.  Let  the  tone 
B  be  a  very  little  higher  than  A.  Then  the  fundamentals  and 
each  pair  of  the  overtones  produce  beats,  and  the  dissonance  is 
great.  As  the  pitch  of  B  is  raised,  this  dissonance  falls  off  as 
each  pair  of  jarring  tones  becomes  more  widely  separated  in 
pitch.  This  falling  off  in  the  dissonance  continues  until  the 
fifth  overtone  of  B  comes  to  be  adjacent  to  the  sixth  overtone 
of  A.  The  jarring  action  of  this  pair  of  tones  then  causes  a  dis- 
tinct rise  in  the  dissonance,  followed  by  a  rapid  fall  as  the  pair 
come  into  unison.  As  the  tone  B  continues  to  rise  in  pitch, 
there  is  a  rapid  rise  in  the  dissonance,  and  so  on.  When  the 

*  The  consonance  of  simple  tones  is  discussed  in  Art.  805. 


MUSICAL   INTERVALS   AND   SCALES. 


183 


C5 


All  of  B 


ELEMENTS   OF   PHYSICS. 

fifth  overtone  of  B  is  in  unison  with  the  sixth  overtone  of  A, 
the  frequency  ratio  of  B  :  A  is  equal  to  6:5. 

The  ordinates  of  the  curve  in  Fig.  554  show  the  values  of 
the  dissonance  of  two  violin  tones  A  and  B,  in  so  far  as  the 
overtones  2,  3,  4,  5,  6,  and  8  are  concerned.  The  fractions 

z? 

below  the  line  BAB  show  the  values  of  the  frequency  ratio  — 

for  the  various  minima  of  dissonance.  This  curve  is  adapted 
from  a  more  complex  one  by  Helmholtz.  The  numerical  evalua- 
tion *  of  dissonance  is  an  approximation. 

Remark.  —  Combination  tones  have  some  action  in  the  pro- 
duction of  dissonance  when  two  tones  are  sounded  together. 
The  dissonance  due  to  combination  tones  passes  through  mini- 
mum values  for  the  same  pitch  intervals  as  does  the  dissonance 
due  to  overtones. 

816.  Consonant  intervals.  —  A  pitch  interval  between  two 
tones,  for  which  the  tones  are  approximately  consonant,  is 
called  a  consonant  interval.  Figure  554  shows  the  various 
consonant  intervals.  The  following  table  exhibits  the  various 
consonant  intervals  in  the  order  of  the  completeness  of  their 
consonance,  together  with  their  names. 

TABLE  OF  CONSONANT  INTERVALS. 

i  :  i  Unison  3 : 5  Major  Sixth 

i  :  2  Octave  4 : 5  Major  Third 

2 : 3  Fifth  5 : 6  Minor  Third 

3 : 4  Fourth  5 : 8  Minor  Sixth 

The  consonance  of  the  octave  is  complete.  That  of  the  fifth 
is  very  nearly  so. 

The  bounding  of  consonant  intervals.  —  Those  overtones  and 
combination  tones  which  determine  a  consonant  interval  by 
their  coincidence,  and  which  produce  the  greater  part  of  the 
dissonance  when  the  interval  is  slightly  out  of  tune,  are  said 
to  bound  the  interval. 

*  See  Helmholtz,  Tonempfindungen,  Beilage  XV. 


MUSICAL   INTERVALS   AND   SCALES.  185 

The  great  increase  in  dissonance,  due  to  a  slight  error  of 
tuning  of  a  consonant  interval,  is  the  basis  for  a  remarkably 
acute  sense  which  we  have  of  the  accuracy  of  these  intervals. 
This  acute  sense  of  pitch  of  consonant  tones  has  a  great  deal 
to  do  with  the  effectiveness  of  consonant  intervals  in  music ; 
for  there  can  be  no  refinement  of  musical  expression  without 
an  acute  sense  to  seize  upon  it,  and  it  is  the  ultimate  depend- 
ence of  this  acute  sense  upon  the  presence  of  prominent  over- 
tones which  explains  the  peculiar  musical  value  of  such  tones 
as  those  of  the  violin  and  of  the  human  voice. 

817.  The  variation  of  the  character  of  the  consonant  inter- 
vals with  timbre.  —  A  consonant  interval  is  the  more  striking 
in  character  in  proportion  as  it  is  more  sharply  bounded. 
Therefore  the  character  of  a  given  consonant  interval  varies 
with  the  timbre  of  the  tones  used;  that  is,  with  the  relative 
loudness  of  the  various  overtones. 

This  is  exemplified  by  clarionet  tones,  which  have  only  odd 
overtones.  All  of  the  consonant  intervals  are  indeed,  in  this 
case,  bounded  by  combination  tones,  but  the  major  sixth  (3 : 5) 
is  more  striking  in  character  than  the  fourth  (3  14),  and  perhaps 
even  as  sharply  defined  as  the  fifth  (2:3);  while  the  minor 
third  (5  :6)  and  the  major  third  (4  :  5)  are  ill  defined.  A  major 
third  (4 : 5),  formed  by  a  violin  tone  and  a  clarionet  tone,  is  very 
much  more  striking  when  the  clarionet  tone  is  the  lower,  so 
that  its  fifth  overtone  coincides  with  the  fourth  overtone  of  the 
violin,  than  it  is  when  the  violin  tone  is  the  lower ;  and  a  minor 
third  (5  : 6)  is  much  more  striking  when  the  violin  tone  is  the 
lower. 

In  the  case  of  pure  tones,  such  as  the  tones  of  tuning  forks 
and  of  wide  organ  pipes  closed  at  one  end,  combination  tones, 
only,  serve  to  bound  the  consonant  intervals.  With  such  tones, 
the  octave  (i  :  2)  is  pretty  sharply  defined,  the  fifth  (2:3)  less 
sharply,  while  the  remaining  intervals  are  scarcely  bounded  at 
all.  Helmholtz,  indeed,  has  found  that  the  sound  of  two  tuning 


1S6  ELEMENTS   OF   PHYSICS. 

forks  is  smooth,  or  consonant,  whatever  the  pitch  interval, 
provided  only  that  this  interval  is  not  so  near  to  the  fifth  (2  : 3) 
or  to  the  octave  (1:2)  as  to  bring  out  the  dissonances  which 
bound  these  two  intervals. 

818.  The  major  and  minor  accords.  —  Three  tones  which 
form  a  consonant  combination  are  called  an  accord,  or  chord. 
Thus  three  tones,  of  which  the  vibration  frequencies  are  as 
4:5:6,  form  an  accord. 

Any  tone  of  an  accord  may  be  replaced  by  its  octave,  or  may  be 
accompanied  by  its  octave,  without  greatly  altering  the  character 
of  the  accord.  This  is  evident  when  we  consider  that  no  new 
overtones  are  introduced  into  the  sound  by  the  octave. 

The  major  accord  and  its  modifications.  —  The  three  tones  of 
which  the  vibration  frequencies  are  as  4:5:6  constitute  what 
is  called  the  major  accord.  By  replacing  the  first  tone  (4)  by  its 
octave  (8),  we  obtain  a  modification  of  this  accord,  and  by  re- 
placing the  third  tone  (6)  by  its  lower  octave  (3),  we  obtain 
another  modification.  The  three  forms  of  the  major  accord  are, 
therefore, 

3         4         5 

456 

5         6        8 

The  minor  accord  and  its  modifications.  — The  three  tones  of 
which  the  vibration  frequencies  are  as  10 : 12  : 15  constitute  what 
is  called  the  minor  accord.  The  interval  between  the  first  two 
tones  is  a  minor  third  (5  : 6),  between  the  last  two  tones  is  a 
major  third  (4  :  5),  and  between  the  first  and  last  the  interval  is 
a  fifth  (2:3);  so  that  the  minor  accord  contains  the  same  con- 
sonant intervals  as  the' major  accord  (4:5:6).  The  modifica- 
tions of  the  minor  accord  are 

10  12  15 

12  15  20 

15  20  24 


MUSICAL  INTERVALS   AND   SCALES. 


I87 


The  primary  forms  of  the  major  and  minor  accords,  viz., 
4:5:6  and  10:  12  :  15,  are  those  in  which  the  three  tones  are 
separated  by  the  smallest  pitch  intervals. 

Difference  in  character  of  major  and  minor  accords.  —  The 
major  and  minor  accords  contain  the  same  consonant  intervals, 
and  the  coincident  overtones  are  identical  in  the  two  cases. 
The  combination  tones,  however,  are  very  different.  The  fol- 
lowing schedule  shows  the  combination  tones  of  the  first  and 
second  orders. 

The  major  accord. 

Primary  tones  456 

First  difference  tones  I          2 

Second  difference  tones  2345 

The  minor  accord. 

Primary  tones  10     12  15 

First  difference  tones         235 

Second  difference  tones  7     8     9     10     12     13 

This  schedule  shows  that  the  difference  tones  of  the  major 
accord  are  exact  duplications,  or  duplications  in  the  lower 
octaves,  of  the  primary  tones.  That  is,  no  foreign  tones  are 
introduced  into  the  major  accord  by  the  combination  tones. 
On  the  other  hand,  some  of  the  difference  tones  of  the  second 
order,  viz.,  7,  8,  9,  and  13,  which  occur  in  the  minor  accord^  are 
dissonant,  and  give  to  this  accord  a  character  very  different 
from  that  of  the  major  accord. 

819.  Musical  scales.  —  The  successive  tones  in  a  melody 
(see  Art.  822),  and  the  simultaneous  tones  in  harmony  (see 
Art.  823),  are  chosen  with  reference  to  their  consonance. 

The  major  scale.  —  Consider  a  given  tone  c\  The  tones 
which  can  be  used  with  c'  with  musical  effect  are  those  desig- 
nated by  e'b,-e',  f ',  g',  a'\>,  and  a'  in  Fig.  554.  Ignoring  the 
tones  e'b  and  a'b,  which  have  low  degrees  of  consonance  with  cf, 


!88  ELEMENTS   OF   PHYSICS. 

we  have  the  following  series  of  musical  tones,  each  of  which  is 
consonant  with  c1: 

Tones  c'        e'    f    g'     a!        c"  } 

Vibration  frequencies     i         f     f      f      f         2  J 

Remark. — The  tone  c1,  with  reference  to  which  a  series  of 
tones  is  selected,  is  called  the  tonic  of  the  series.  The  tone  g' , 
having  next  to  the  octave  the  most  complete  consonance  with  c' , 
is  called  the  dominant ;  and  the  tone  f',  which  is  next  in  order 
of  consonance,  is  called  the  subdominant  of  the  series. 

For  purposes  of  harmony,  it  is  desirable  to  be  able  to  build 
major  accords  (4:5:6)  upon  the  tonic,  upon  the  dominant,  and 
upon  the  subdominant  of  a  series  of  musical  tones.  Two  of 
these  major  accords  may  be  built  up  with  the  tones  in  series  L, 
namely,  c',  e'y  gf  (4:5:6)  and  /',  a',  c"  (4:5:6).  To  build  a 
major  accord  upon  g1 ,  two  additional  tones,  say  b1  and  d'} ',  are 
required  such  that  g' :  b'  :  dn  =  4  :  5  :  6.  Therefore  the  vibra- 
tion frequencies  of  b1  and  d"  are  J^-  and  |-  respectively.  Taking 
a  tone  d} ',  an  octave  below  d" ,  we  have  the  series  : 

Tones  c'    d'     e'    f    g'     a'     b'    c"  \ 

Vibration  frequencies       i      f      f      f      f      f    -1/-    2  J 

This  is  the  ordinary  musical  scale,  called  the  major  scale. 

The  minor  scale.  —  Choosing,  with  the  help  of  Fig.  554,  the 
tones  below  c',  which  are  most  nearly  consonant  with  c't  we 
have  the  series  : 

Tones  c        fo   f    g    d&        c' 

Vibration  frequencies     \         fill         I 

or,  in  order  that  this  series  may  be  more  easily  compared  with 
L,  we  may  choose  all  of  these  tones  an  octave  higher,  whence 
we  obtain  the  following  series  of  musical  tones : 

Tones  c'        e>\>   f    g'     a'b        c" 

Vibration  frequencies     i          III!  2 

Remark. — This  series  III.  is  more  melodious  when  sounded 
in  the  order  of  descending  pitch  than  when  sounded  in  the 
reverse  order,  for  the  reason  that  the  tones  of  the  series  are 


MUSICAL   INTERVALS   AND   SCALES.  189 

more  nearly  consonant  with  c"  than  with  c',  and  whichever 
of  these  tones  is  sounded  first  is  made  correspondingly  promi- 
nent. The  series  I.  (and  also  II.)  is  more  melodious  when 
sounded  in  the  order  of  ascending  pitch. 

The  series  III.  includes  the  two  minor  accords  (10,  12,  15) 
cf»  e^>  g  '>  and  f  y  &rb,  c"  .  To  build  a  minor  accord  upon  g1  , 
two  additional  tones,  say  tf\>  and  du  ',  are  required,  such  that 
g'  :  b'\>\  d"  —  10  :  12  :  15,  so  that  the  vibration  frequency  of 
b'\>  is  f  and  the  vibration  frequency  of  dn  is  |-.  Taking  a  tone 
dl  r,  an  octave  below  d",  we  have  the  series  : 

Tones  c'     d1     e'\>   f    g'     afy    b'\>    c"  }  Jy 

Vibration  frequencies     iffff       I       1       2  J 

This  series  of  tones  is  called  the  descending  minor  scale. 
For  purposes  of  melody,  this  scale  is  changed  to  the  following 
for  ascending  movements  : 

Tones  c'     d'     e'\>   f    g'     a!    V    c"  j  y 

Vibration  frequencies    i      f      f      f      f      f    -1/-     2  J 

This  is  called  the  ascending  minor  scale. 

Remark.  —  The  tones  which  are  consonant  with  the  tonic  c1 
are  called  related  tones  of  the  first  order.  The  tones  d'  and  b' 
of  the  major  scale  and  d'  and  b'b  of  the  minor  scale  which  are 
consonant  with  g'  are  called  related  tones  of  the  second  order 
(that  is,  related  to  the  tonic  c'). 

The  scales  II.  and  IV.  are  better  suited  to  the  require- 
ments of  harmony  than  is  scale  V.  The  scale  II.  is  suited 
to  harmony  in  which  major  accords  predominate.  It  is  for 
this  reason  called  the  major  scale.  The  scale  IV.  is  suited  to 
harmony  in  which  minor  accords  predominate.  It  is  for  this 
reason  called  the  minor  scale. 

The  following  schedules  exhibit  all  of  the  major  and  minor 
accords  which  can  be  formed  of  the  tones  of  the  major  and 
minor  scales. 

The  major  scale. 

major  accord    major  accord    major  accord 

0    cf~   e'   ^'   V    ~d" 


minor  accord    minor  accord 


o  ELEMENTS    OF   PHYSICS. 

The  minor  scale. 

major  accord      major  accord 


/    aft    c'    e'\>    g1   b'\)    d1 

minor  accord      minor  accord     minor  accord 

The  tonic  accord  is  shown  in  each  case  by  the  bold-faced 
type.  In  the  major  scale  the  tonic  accord,  the  dominant 
accord,  and  the  subdominant  accord  are  major  accords.  In  the 
minor  scale  these  accords  are  minor  accords. 

The  naming  of  consonant  intervals.  —  The  intervals  between 
the  tonic  and  the  third  and  sixth  tones  of  the  major  scale  are 
called  the  major  third  and  major  sixth  respectively.  The  inter- 
vals between  the  tonic  and  the  third  and  sixth  tones  of  the 
minor  scale  (IV)  are  called  the  minor  third  and  minor  sixth 
respectively.  The  intervals  between  the  tonic  and  the  fourth 
and  fifth  tones  of  either  scale  are  called  the  fourth  and  fifth 
respectively. 

820.  Musical   expression.  —  Music   is   said  to   be  expressive 
when  it  appeals  in  a  distinct   manner  to  the  emotions.     The 
primary  forms  of  musical  expression  are  rhythm,  melody,  har- 
mony, and   modulation.      The   artistic  use  of  these   elements 
is  largely  traditional  and  conventional ;  still  the  development  of 
musical  method  has  been  largely  determined  by  physical  facts 
or  laws.     The  following  articles  give  brief  statements  of  the 
essential  nature  of  rhythm,  melody,  harmony,  and  modulation. 

821.  Rhythm.  —  The  rapidity  of  succession  of  the  tones  in 
music  and  the  manner  in  which  successive  tones  are  set  off 
in  groups  is  called  rhythm. 

The  rhythmic  grouping  of  tones  enables  a  listener  to  appre- 
hend them  more  clearly,  and  therefore  permits  the  effective 
use  of  more  extended  phrases  in  melody  and  modulation  than 
would  otherwise  be  possible.  Rhythm  is  also  effective  in 
giving  expression  to  vigor  or  languor  according  to  the  rapidity 
of  the  movement ;  and  changes  of  rhythm  serve  to  mark  the 


MUSICAL   INTERVALS   AND   SCALES.  191 

progress  of  long  compositions,  and  at  the  same  time  to  give 
variety. 

822.  Melody. — A  sequence  of  tones  is  called  a  melody.    The 
shades  of  expression  produced  by  different  sequences  of  tones 
depend  mainly  upon  the  coincidence  of  overtones  of  successive 
notes.      When  two  successive   notes   have   several   coincident 
overtones,   the   progressive    effect   of  the   melody,  or,  as  it  is 
technically  called,  the  movement  comes  to  a  momentary  stop 
of  a  more  or  less  decided  character.     Thus  a  repetition  of  the 
same  tone  is  a  full  stop  in  the  movement,  and  the  progressive 
effect  becomes  more  and  more  distinct  as  the  consonant  rela- 
tion of  successive  notes  becomes  less  and  less  marked.     When 
successive  tones  are  dissonant,  the  movement  is  abrupt,  and  is 
expressive  of  sudden  emotions,  such  as  surprise.     The  interval 
if  between  b'  and  c"  of  the  major  scale  is  an  exception  to  this 
last  statement.     The  note  b'  seems  to  serve  as  a  catch  note,  or 
a  threshold,  to  be  passed  in  reaching  c" . 

823.  Harmony.  —  The  simultaneous  use  of  a  number  of  tones 
in  music  is  called  harmony.     The  shades  of  musical  expression 
produced   by   different   combinations   of   tones  depend  mainly 
upon  the  degree  of  consonance  of  the  combination.     Harmony 
built  upon  major  accords  is  strikingly  different  in  expression 
from  that  built  upon  minor  accords.     The  first  gives  an  expres- 
sion of  contentment  and  cheerfulness,  while  the  latter  is  expres- 
sive of  discontent  and  sadness. 

824.  Modulation. —  Two  accords  are  said  to  be  related  when 
they  have  a  common  tone  or  two  common  tones.     A  sequence 
of  accords,  each  of  which  is  related  to  the  one  preceding  it,  is 
called   a   modulation.      The   possibility   of   modulation   in   the 
major  and  minor  scales  are  exhibited  by  the  following  schedule, 
taken  from  Art.  819. 

Major  scale. 

major  accord  major  accord  major  accord 


/   a   c'   e'   g'   b'   d" 

minor  accord  minor  accord 


ICJ2  ELEMENTS   OF   PHYSICS. 

Minor  scale. 

major  accord  major  accord 

c   e'fr 


minor  accord  minor  accord  minor  accord 

More  extended  modulations  than  those  exhibited  in  this 
schedule  require  the  use  of  tones  related,  in  the  third  order, 
to  the  tonic  c'.  Let  us  consider,  for  example,  an  extension  in 
both  directions  of  the  modulation  of  the  major  scale.  We  have: 


iv      x     f      a      c'      e'     g1      b'      d"     y       z 

The  brackets  represent  major  accords.  The  tones  y  and  z 
are  not  related  to  c',  but  they  are  related  to  g'  exactly  as  b'  and 
d"  respectively  are  related  to  c'.  Thus  the  extension  of  this 
modulation  upwards  leads  to  a  set  of  tones  having  a  new  tonic, 
namely,  g* .  In  like  manner  the  tones  w  and  x  are  related  to 
/  exactly  as  f  and  a  respectively  are  related  to  c',  so  that  an 
extension  of  this  modulation  downwards  leads  to  a  set  of  tones 
having  a  new  tonic,  namely,/  Such  an  extended  modulation 
in  one  direction  or  the  other  is  called  a  change  of  key. 

In  popular  music,  for  example  dance  music,  the  modulation 
is  confined  to  the  tonic,  dominant,  and,  subdominant  accords, 
which  succeed  each  other  periodically.  The  greater  musicians 
often  use  very  extended  and  very  complex  modulations,  making 
use  of  major  and  minor  accords  in  one  and  the  same  phrase, 
with  occasional  dissonances  for  the  sake  of  contrast. 

825.  The  tempered  scale.  —  A  great  number  of  distinct  tones 
is  required  for  extended  modulation,  and  it  would  be  impracti- 
cable for  a  player  to  use  a  piano  or  an  organ  having  a  separate 
key  (and  string  or  organ  pipe)  for  each  tone.  This  difficulty  is 
overcome,  at  the  expense  of  accuracy  of  tuning  of  the  various 
consonant  intervals,  by  the  use  of  what  is  called  the  tempered 
scale.  This  scale  consists  of  twelve  tones  (thirteen,  counting 
both  end  tones)  in  each  octave,  the  pitch  intervals  between 
successive  tones  being  equal.  The  octave  is  thus  divided  into 


MUSICAL   INTERVALS   AND   SCALES. 


193 


twelve  equal  pitch  intervals.  The  logarithm  of  the  frequency 
ratio  of  each  of  these  intervals  is  therefore  one-twelfth  of  the 
logarithm  of  2,  which  is  the  frequency  ratio  of  the  octave. 
(Compare  Art.  814.)  The  following  table  shows  the  logarithms 
of  the  frequency  ratios  of  each  tone  of  the  major  scale  to  the 
tonic,  and  also  the  logarithms  of  the  frequency  ratios  of  each 
tone  of  the  tempered  scale  to  the  tonic. 


Tonic. 

Number  of  tone  in 

tempered  scale 

I 

2 

3 

4 

5 

6 

7 

Log  of  ratio  of  tone 

of  tempered  scale 

to  the  tonic 

0 

.0251 

.0502 

•0753 

.1003 

.1254 

.1505 

Nearest      tone      in 

major  scale 

c' 

d' 

e' 

/ 

Log  of  ratio  of  tone 

of  major  scale  to 

tonic 

0 

.0511 

.0979 

.1250 

Difference  of  logs 

0 

—  .0009 

+  .0024 

+  .0004 

Octave. 

Number  of  tone  in 

tempered  scale 

8 

9 

10 

ii 

12 

13 

Log  of  ratio  of  tone 

of  tempered  scale 

to  the  tonic 

.1756 

.2007 

.2258 

.2509 

.2759 

.3010 

Nearest      tone      in 

major  scale 

f 

a' 

V 

c" 

Log  of  ratio  of  tone 

of  major  scale  to 

tonic 

.1761 

.2219 

.2730 

.3010 

Difference  of  logs 

-  .0005 

-f  .0039 

+  .0029 

o 

194 


ELEMENTS   OF   PHYSICS. 


This  table  shows  that  the  first,  third,  fifth,  sixth,  eighth, 
tenth,  twelfth,  and  thirteenth  tones  of  the  tempered  scale  are 
very  nearly  in  unison  with  the  successive  tones  of  the  major 
scale.  These  tones  may,  in  fact,  be  used  for  the  tones  of  the 
major  scale ;  and  since  the  intervals  of  the  tempered  scale  are 
all  equal,  it  is  clear  that  any  tone  of  the  tempered  scale  may  be 
chosen  as  a  tonic,  and  that  the  third,  fifth,  sixth,  eighth,  tenth, 
twelfth,  and  thirteenth  tones,  counting  from  the  chosen  one,  con- 
stitute a  major  scale.  All  such  major  scales  are  equally  well  in 
tune.  Indefinite  modulations  may  be  carried  out  on  this  scale 
inasmuch  as  any  tone  reached  in  a  modulation  has  a  group  of 
tones  related  to  it  as  a  tonic.  The  minor  scale  may  be  made 
up  also  from  the  tempered  scale  and  indefinite  modulations 
may  be  performed. 


OF  THB 

DIVERSITY 


^       -1 I  VER! 

V  •',- , 


INDEX. 


NUMBERS    RELATE  TO   PAGE. 


ABBE'S  LAW,  57. 
Aberration,  chromatic,  58. 

spherical,  43,  56. 
Absorption,  coefficients  of,  144. 

selective,  138. 

Accommodation  of  the  eye,  65. 
Accords,  major  and  minor,  186. 
Achromatic  lenses,  58,  74. 
Air  columns,  vibration  of,  1 54. 
Amplitude  of  vibration,  148. 

of  a  wave  train,  13. 
Anastigmatic  lens  systems,  63. 
Angle,  the  polarizing,  127. 

visual,  65. 

of  incidence  defined,  37. 

of  a  lens,  56. 

of  refraction  defined,  37. 
Antinodes,  defined,  15. 
Aperture,  defined,  55. 
Aplanatic  points,  57. 

surface,  defined,  43. 
Aqueous  humor,  64. 
Astigmatic  pencils,  24. 
Astigmatism  of  lenses,  59. 
Audibility,  limits  of,  150. 
Auditory  nerves,  2. 
Angstrom  units,  99. 

BASILAR  MEMBRANE,  THE,  174. 
Beats,  nature  of,  175. 

the  method  of,  150. 

Becquerel  on  radiation  from  uranium,  136. 
Becquerel  rays,  the,  146. 
Bell,  clang  of  a  Russian,  163. 
Bells,  overtones  of,  163. 
Black  bodies,  properties  of,  140. 


Bouguer's  principle,  116. 

Bravais  and  Martins,  velocity  of  sound,  5. 

Brewster's  magnifier,  61. 

Bright  line  spectra,  77. 

Brightness  and  color,  101. 

distribution  of,  1 20. 

intrinsic,  115. 

standards  of,  112. 
Bunsen's  photometer,  117. 

CAMERA,  THE  PHOTOGRAPHIC,  66. 
Candles,  British  and  German,  113. 
Carcel  lamp,  the,  1 14. 
Caustic,  defined,  30. 
Change  of  phase  by  reflection,  17. 
Chemical  effects  of  radiation,  136. 
Chladni's  figures,  164. 
Chords,  major  and  minor,  1 86. 
Chromatic  aberration,  58. 
Circular  polarization,  126. 
Clang,  analysis  of,  169. 

defined,  153. 

of  a  Russian  bell,  163. 
Clarionet,  the,  158. 
Cochlea,  the,  174. 
Coefficients  of  absorption,  144. 
Collinator,  76. 
Color  and  brightness,  101. 
Color  blindness,  105. 

testing  for,  1 10. 
Color  by  absorption,  141. 

by  interference,  103. 

by  selective  radiation,  103. 

by  selective  reflection  and  transmis- 
sion, 103. 

mixing,  104. 


'95 


196 


ELEMENTS    OF   PHYSICS. 


Color  sensations,  the  primary,  105. 

the  Young-Helmholtz  theory  of,  105. 

top,  105. 
Colors  due  to  homogeneous  light,  102. 

due  to  mixed  light,  102. 

of  thin  plates,  86. 

saturated,  107. 

surface,  141. 
Combination  tones,  176. 
Composition  of  light,  how  measured,  103. 
Concave  and  convex  mirrors,  31. 
Concave  gratings,  99. 
Conjugate  foci,  of  a  lens,  47. 

of  a  mirror,  32. 

out  of  axis,  34. 

of  a  lens  system,  50. 

planes,  34. 

points  out  of  axis,  48. 
Conjugates,  geometrical  construction  for, 

49. 

Consonance  of  tones,  159,  182. 
Consonant  intervals,  184. 
Continuous  spectra,  77. 
Contrast  effects,  107. 
Convergent  lenses,  45. 
Cornea,  the,  64. 
Cornet,  the,  159. 
Cornu's  measurement  of  the  velocity  of 

light,  7. 

Corpuscular  theory  of  light,  2. 
Correction  for  spherical  aberration,  56. 
Corrections,  simultaneous,  of  lenses,  61. 
Crehore  and  Squier,  on  the  photochrono- 

graph,  134. 

Crova's  method  in  photometry,  123. 
Crystals,  axial  and  biaxial,  130. 
Curvature  of  field,  61. 

DAMPING  OF  VIBRATIONS,  167. 
Dark  line  spectra,  78. 
Daylight  and  gaslight  compared,  103. 
Diaphragms,  motion  of,  165. 
Dichroic  vision,  104. 

peculiarities  of,  108. 
Difference  tones,  177. 
Diffraction,  defined,  88. 

gratings,  94. 


Diffraction  past  an  edge,  89. 

past  the  edges  of  a  narrow  strip,  93. 

through  a  slit,  91. 

of  sound  and  the  sense  of  direction, 

179. 

Direct  vision  spectroscopes,  74. 
Direction  of  sounds,  the  sense  of,  179. 
Discord  and  beats,  176. 
Dispersion,  the  Helmholtz  theory  of,  144. 

resonant,  145. 

Displacement  of  fringes,  83. 
Distortion  of  lenses,  60. 
Distribution  of  brightness,  1 20. 
Divergent  lenses,  45. 
Double  refraction,  128. 
Doublet,  defined,  49. 

Huygens',  62. 

Ramsden's,  62. 

Wollaston's,  62. 
Doublets,  aplanatic,  57. 

symmetrical,  60. 

EAR,  THE,  173. 
Echo,  the,  177. 
Elliptical  mirrors,  31. 
polarization,  126. 
Emission,  selective,  138. 
Emission    and    absorption,   equality   of, 

137- 
Energy,  distribution  of,  in  spectrum,  144. 

radiant,  135. 

stream  in  a  wave  train,  13. 
End  organs,  I. 
Ether,  the  luminiferous,  3. 
Exchanges,  the  principle  of,  136. 
Expression,  musical,  190. 
Extraordinary  ray,  the,  129. 
Eye,  the,  64. 
Eyepieces  used  in  practice,  61. 

FARSIGHTEDNESS,  65. 
Field,  angle,  of  lenses,  56. 

curvature  and  flatness  of,  61. 
Fizeau   and  Cornu   on   the   velocity    of 

light,  7. 

Flatness  of  field,  61. 
Flicker  photometer,  the,  124. 


INDEX. 


197 


Fluorescence  and  phosphorescence,  146. 
Focal  length  of  a  mirror,  33. 
Foci,  virtual  and  real,  33. 

of  a  lens,  47. 

of  a  mirror,  32. 
Foucault,  velocity  of  light,  8. 
Fourier's  theorem,  18. 

applied  to  vibration,  148. 
Fraunhofer  lines,  78. 
Frequency  of  a  wave  motion,  12. 

of  vibration,  defined,  147. 
Fringes,  displacement  of,  83. 
Fundamental  tone,  of  a  pipe,  155. 

of  a  string,  160. 

GASLIGHT    AND    DAYLIGHT    COMPARED, 

103. 

German  candles,  113. 
Grating,  the  concave,  99. 

the  diffraction,  94. 

Greeley,  velocity  of  sound  at  low  temper- 
»          atures,  4. 

HALF-PERIOD  ZONES,  21,  88. 

Harmonic  overtones,  153. 

Harmony  and  modulation,  191. 

Hastings'  magnifier,  62. 

Hearing  at  a  distance,  influence  of  wind 

and  temperature  on,  178. 
Heat,  radiant,  135. 
Hefner  lamp,  the,  115. 
Helmholtz's  theory  of  dispersion,  144. 
Holmgren  test  for  color  blindness,  1 10. 
Homocentric  pencils,  24. 
Homogeneous  light,  73. 

color  due  to,  102. 

luminosity  of,  IOI. 

Huygens'    construction    for   wave  front, 
20. 

doublet,  62. 

principle,  19. 

principle    applied    to   reflection   and 
refraction,  26. 

theory  of  double  refraction,  129. 

ILLUMINATION,  INTENSITY  OF,  115. 
Image  in  a  plane  mirror,  29. 


Images,  distortion  of,  60. 

magnification  of,  35. 

in  spherical  mirrors,  34. 
Incidence,  angle  of,  37. 
Index  of  refraction,  37. 
Infra-red  rays,  73. 
Interference,  color  by,  103. 

from  similar  sources,  82. 

of  sound,  175. 

fringes,  defined,  83. 

Lloyd's  mirror  for,  86. 

Newton's  arrangement  for,  85. 
Intervals  of  pitch,  181. 
Inverse  squares,  law  of,  112. 

KIRCHHOFF'S  LAW,  137. 

Kirchhoff  and  Bunsen  on  reversal  of  lines, 

79- 

Koenig  on  color  curves,  107. 
Krigar-Menzel   and    Raps'    experiment, 

163. 
Kundt's  experiment,  160. 

LABYRINTH  OF  THE  EAR,  174. 
Langley's  bolometer,  142. 
Lantern,  the  magic,  66. 
Law  of  aplanatism  (Abbe),  57. 

of  Kirchhoff,  137. 

of  normal  radiation,  137. 

of  Snell,  37. 

of  inverse  squares,  112. 
Lens,  foci  of  a,  47. 

of  the  eye,  64. 
Lens  systems,  anastigmatic,  63. 

defined,  49. 

foci  of,  51. 

inverse  principal  planes  of,  52. 

nodal  points  of,  53. 

principal  planes  of,  51. 

specification  of,  50. 
Lenses,  achromatic,  38,  74. 

distortion  of,  60. 

diverging  and  converging,  45. 

field  angle  of,  56. 

numerical  aperture  of,  55. 

orthoscopic,  60. 

rectilinear,  60. 


198 


ELEMENTS   OF   PHYSICS. 


Light,  corpuscular  theory  of,  2. 

homogeneous  or  monochromatic,  73. 

the  sensation  of,  2. 

standards,  112. 

velocity  of,  7. 

wave  theory  of,  3. 

white,  defined,  IO2. 
Limits  of  audibility,  1 50. 
Lissajous'  figures,  152. 
Lloyd's  mirrors,  86. 
Longitudinal  transverse  waves,  9. 
Loudness  of  sound,  150. 
Luminescence,  146. 
Luminiferous  ether,  3. 
Luminosity,  curve  of,   for  the  red-blind 

eye,  109. 

Luminosity  of  homogeneous  light,  101. 
Lummer-Brodhum  photometer,  119. 

MAGIC  LANTERN,  66. 
Magnification  of  images,  35. 
Magnifying  glasses,  61,  67. 
Magnifying  power,  defined,  67. 

of  microscopes,  68. 
Major  and  minor  accords,  186. 
Major  scale,  the,  189. 
Manometric  flames,  165. 
Measurement  of  radiant  heat,  142. 
Melody  and  harmony,  191. 
Membrane,  the  basilar,  174. 
Methven  screen,  the,  114. 
Michelson's  measurement  of  the  velocity 

of  light,  8. 
Micron,  the,  99. 
Microscope  objectives,  62. 
Microscopes,  compound,  68. 

magnifying  power  of,  68. 

simple,  67. 

Minimum  deviation,  38. 
Minor  accords,  186. 
Minor  scale,  the,  190. 
Mirror,  foci  of  a,  32. 
Mirrors,  concave  and  convex,  31. 

elliptical,  31. 

images  in  convex,  34. 

parabolic,  31. 

plane  and  spherical,  29. 


Modulation,  191. 
Monochromatic  light,  73. 
Musical  expression,  190. 

scales,  187. 

tones  defined,  148. 

NEARSIGHTEDNESS,  65. 
Nerves,  the  auditory,  2. 

the  optic,  i. 

the  sensory,  I. 
Newcomb's  measurement  of  the  velocity 

of  light,  8. 

Newton,  seven  spectrum  colors  of,  102. 
Newton's  rings,  88. 
Nichols  radiometer,  the,  143. 

spectrophotometer,  the,  80. 
Nicol  prism,  the,  131. 
Nicol  prisms  in  photometry,  122. 
Nodal  points  of  a  lens  system,  53. 
Nodes,  defined,  14. 
Noises,  defined,  149. 
Normal  and  prismatic  spectra,  80. 
Normal  radiation,  the  law  of,  137. 

OBJECTIVES  FOR  MICROSCOPES,  62. 

for  telescopes,  63. 

photographic,  63. 
Opaque  bodies,  defined,  112. 
Opera  glasses,  72. 
Optic  nerves,  I,  64. 
Ordinary  and  extraordinary  rays,  129. 
Organ  pipes,  157. 
Orthoscopic  lenses,  60. 
Overtones,  defined,  153. 

PARABOLIC  MIRRORS,  31. 

Pencils,  homocentric  and  astigmatic,  24. 

Penumbra,  the,  23. 

Period  of  a  wave  motion,  12. 

of  a  vibration,  148. 
Phase,  change  of,  by  reflection,  17. 

of  vibration,  148. 

of  a  wave  train,  13. 
Phonautograph,  the,  172. 
Phonograph,  the,  171. 
Phosphorescence  and  fluorescence,  146. 
Photographic  camera,  66. 
Photographic  objectives,  63. 


INDEX. 


199 


Photometer,  the  Bunsen,  117. 

the  flicker,  124. 

the  Lummer-Brodhun,  119. 

the  Rumford,  117. 

the  shadow,  1 1 7. 
Photometry,  simple,  116. 

of  light  differing  in  composition,  122. 

Whitman's  method,  124. 
Pipe,  the  organ,  157. 
Pitch,  defined,  150. 

influence  of  relative  motion  on,  179. 

intervals,  181. 

-measurement  of,  150. 
Plane  of  polarization,  127. 

of  polarization,  the  rotation  of,  133. 

waves,  refraction  of,  36. 
Planes,  conjugate,  34. 
Polariscope,  the,  132. 
Polarization,  defined,  125. 

circular  and  elliptical,  126. 

by  double  refraction,  131. 

by  reflection,  126. 

by  tourmaline,  126. 
Polarizing  angle,  the,  127. 
Potassium  chromate,  color  of,  104. 
Prevost's  principle  of  exchanges,  136. 
Primary  color  sensations,  105. 
Principal  foci  of  a  lens  system,  51. 
Principal  focus  of  a  lens,  45. 

of  a  mirror,  32. 

Principal  planes  of  a  lens  system,  51. 
Principle  of  exchanges,  the,  136. 
Prism,  defined,  38. 

the  Nicol,  131. 
Proper  stimuli,  i. 

RADIANT  ENERGY,  135. 
Radiant  heat,  135. 

luminous  and  chemical  effects  of,  136. 

measurement  of,  142. 
Radiation  in  a  closed  system,  137. 

color  by,  103. 

the  law  of  normal,  137. 
Radiometer,  the,  143. 
Ramsden's  doublet,  62. 
Ray,  the  ordinary,  128. 
Rays,  defined,  22. 


Real  and  virtual  foci,  33. 
Rectilinear  lenses,  60. 
Reed  pipes,  158. 

Reflection,  application  of  Huygens'  prin- 
ciple to,  26. 

color  by,  103. 

from  a  plane  surface,  27. 

polarization  by,  126. 

regular  and  diffuse,  26. 

selective,  138. 

total,  41. 

with  and  without   change  of  phase, 

17- 

Refraction,  defined,  26, 
angle  of,  37. 
application  of  Huygens'  principle  to, 

26. 

double,  128. 
at  spherical  surfaces,  42. 
of  a  plane  wave,  36. 
of  spherical  waves,  38. 
Refractive  index,  37. 
Regnault's  measurement  of  the  velocity 

of  sound,  5. 

Resonance  and  resonators,  168. 
Resonant  dispersion,  145. 
Retina,  the,  64. 
Reversal  of  sodium  lines,  79. 
Rhythm,  190. 
Rods  and  strings,  longitudinal  vibrations 

of,  159- 
Roemer's  measurement  of  the  velocity  of 

light,  7. 

Rotation  of  the  plane  of  polarization,  133. 
Rubens  and  Nichols,  on  radiation  of  long 

waves,  135. 

Rumford's  photometer,  116. 
Russian  bell,  clang  of,  163. 

SACCHARIMETER,  THE,  133. 
Scale,  the  tempered,  192. 
Scales,  major  and  minor,  189. 

musical,  187. 
Segments,  defined,  15. 
Selective  absorption,  138. 

emission,  138. 

reflection,  138. 


200 


ELEMENTS   OF   PHYSICS. 


Selective  transmission,  138. 
Sensations,  defined,  I. 

of  light,  2. 

of  sound,  2. 
Sensory  nerves,  I. 
Shadow,  the  geometrical,  23. 

photometers,  116. 
Shadows,  defined,  23. 
Sharp  and  Turnbull  on  light  standards, 

114. 

Siren,  the,  150. 
Snell's  law,  37. 

Snow,  on  lines  in  the  infra-red,  78. 
Solar  spectrum,  the,  78. 
Sound  rays,  22. 

fringes,  86. 

interference  of,  175. 

loudness  of,  150. 

the  sensation  of,  2. 

persistence  of,  175. 
Sound  shadows,  23. 
Sound,  velocity  in  air,  4. 

velocity  at  low  temperatures,  4. 

velocity  at  high  and  low  levels,  5. 

velocity  in  metals,  5. 

velocity  in  water  and  glass,  6. 

wave  theory  of,  3. 
Sounding  boards,  use  of,  163. 
Specification  of  a  lens  system,  51. 
Speech,  reproduction  of,  171. 
Spectra,  bright  line,  77. 

continuous,  77. 

dark  line,  78. 

normal  and  prismatic,  80. 
Spectrometer,  the  grating,  97. 
Spectrometers,  79. 
Spectroscopes,  described,  76. 
Spectro-photometers,  80. 
Spectro-photometry,  Crova's  method  in, 
123. 

method  of  the  Vierordt  slit,  122. 
Spectroscope,  the  direct  vision,  74. 
Spectrum,  the,  73. 

appearance  of,  to  color-blind  observ- 
ers, 109. 

energy,  curves  of  the,  143. 

the  solar,  78. 


Spherical  aberration,  43,  56. 

mirrors,  29. 

waves,  refraction  of,  38. 
Spinney  ;   curves  of  wave  motions,  18. 
Spy  glasses,  72. 
Standards  of  brightness,  112. 
Stationary  wave  trains,  14. 
Stevens  and  Mayer  on  sound  fringes,  86. 
Stimulus,  defined,  I. 
Strings  and  rods,  longitudinal  vibrations 

of,  159. 

Strings,  simple  and  compound,  vibrations 
of,  162. 

transverse  vibration  of,  160. 
Summation  tones,  177. 
Sun,  the  spectrum  of,  78. 
Superposition,  the  principle  of,  14. 
Surface  color,  141. 
Systems  of  lenses,  49. 

TELESCOPES,  70. 

magnifying  power  of,  71. 
Tempered  scale,  the,  192. 
Thin  plates,  colors  of,  86. 
Timbre,  defined,  152. 
Tones,  combination,  176. 

consonance  of,  182. 

musical,  148. 

summation  and  difference,  177. 
Total  reflection,  41. 
Tourmaline,  optical  behavior  of>  126. 
Transmission,  color  by,  103. 

selective,  138. 

Transparent  and  translucent  bodies  de- 
fined, 112. 
Transverse  and  longitudinal  waves,  9. 

vibrations  of  strings,  160. 
Trichroic  vision,  104. 
Triplet,  the,  defined,  49. 
Tuning  forks,  the  motion  of,  164. 

overtones  of,  163. 
Turnbull  and  Sharp  on  light  standards, 

114. 
Tympanic  membrane,  the,  174. 

ULTRAMARINE  BLUE,  COLOR  OF,  104. 
Ultra-violet  rays,  73. 


INDEX. 


201 


Umbra  and  penumbra,  23. 
Unit,  the  Angstrom,  99. 

VELOCITY  OF  LARGE  AND  SMALL  WATER 
WAVES,  1 80. 

of  light,  7. 

of  sound,  4. 

of     sound     (independent     of    wave 
length),  1 80. 

of  sound  in  water  and  glass,  6. 

of  sound  in  metals,  6. 
Vibrating  segments,  defined,  15. 
Vibration,  amplitude  and  phase  of,  148. 

of  air  columns,  154. 

of  a  particle,  147. 

period  of,  148. 
Vibrations,  damping  of,  167. 

proper  and  impressed,  167. 

simple  and  compound,  147,  148. 

of  rods  and  strings  (longitudinal),  159. 

of  rods  and  plates  (transverse),  183. 
Vibrations,  of  strings  (simple  and  com- 
pound), 162. 
Vierordt's  slit,  122. 
Violle  standard  of  light,  the,  115. 
Virtual  and  real  foci,  33. 
Vision,  dichroic  and  trichroic,  104. 

peculiarities  of  dichroic,  108. 


Visual  angle,  65. 
Vitreous  humor,  the,  64. 
Vocal  organs,  the,  159. 
Vowel  sounds  analyzed,  169. 
how  produced,  159. 

WATER  WAVES,  VELOCITIES  OF,  180. 
Wave  front,  defined,  19. 

Huygens'  construction  of,  2O. 
Wave  length,  defined,  12. 

measurement  of,  97. 
Wave  motion,  equations  of,  10. 
Wave  theory  of  light,  3. 
Wave  trains,  12. 

simple  and  compound,  18. 

stationary,  14. 
Waves,  nature  of,  9. 
White  bodies,  properties  of,  140. 

light,  defined,  102. 
Whitman's  flicker  photometer,  124. 
Wollaston's  doublet,  62. 
Word,  manometric  flame-image  of,  166. 

YELLOW  SPOT,  THE,  65. 
Young-Helmholtz   theory  of  color,  the, 
105. 

ZONE  PLATES,  93. 

Zones,  half-period,  21,  88. 


THE  ELEMENTS  OF  PHYSICS. 


BY 


EDWARD  L.   NICHOLS,   B.S.,   Ph.D., 

Professor  of  Physics  m  Cornell  University, 


AND 


WILLIAM  S.   FRANKLIN,   M.S., 

Professor  of  Physics  and  Electrical  Engineering  at  the  Iowa  Agricultural  College,  Ames,  la. 
WITH    NUMEROUS    ILLUSTRATIONS. 


(Vol.      I. 

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laboratory,  together  with  demonstrations  and  elementary  statements  of  principles.  It  is 
assumed  that  the  student  possesses  some  knowledge  of  analytical  geometry  and  of  the  cal- 
culus. In  the  second  volume  more  is  left  to  the  individual  effort  and  to  the  maturer  intel- 
ligence of  the  practicant. 

A  large  proportion  of  the  students  for  whom  primarily  this  Manual  is  intended,  are  pre- 
paring to  become  engineers,  and  especial  attention  has  been  devoted  to  the  needs  of  that 
class  of  readers.  In  Vol.  II.,  especially,  a  considerable  amount  of  work  in  applied  elec- 
tricity, in  photometry,  and  in  heat  has  been  introduced. 

COMMENTS. 

"  The  work  as  a  whole  cannot  be  too  highly  commended.  Its  brief  outlines  of  the 
various  experiments  are  very  satisfactory,  its  descriptions  of  apparatus  are  excellent;  its 
numerous  suggestions  are  calculated  to  develop  the  thinking  and  reasoning  powers  of  the 
student.  The  diagrams  are  carefully  prepared,  and  its  frequent  citations  of  original 
sources  of  information  are  of  the  greatest  value."  —  Street  Railway  Journal. 

"  The  work  is  clearly  and  concisely  written,  the  fact  that  it  is  edited  by  Professor  Nichols 
being  a  sufficient  guarantee  of  merit."  —  Electrical  Engineering. 

"  It  will  be  a  great  aid  to  students.  The  notes  of  experiments  and  problems  reveal 
much  original  work,  and  the  book  will  be  sure  to  commend  itself  to  instructors." 

—  San  Francisco  Chronicle. 


THE    MACMILLAN    COMPANY, 

NEW  YORK:  CHICAGO: 

66  FIFTH  AVENUE.  ROOM  23,  AUDITORIUM. 


A  LABORATORY  MANUAL 


OF 


EXPERIMENTAL    PHYSICS, 

BY 

W.  J.  LOUDON  and  J.   C.  McLE*NNAN, 

Demonstrators  in  Physics,  University  of  Toronto, 
Cloth.    8vo.     pp.  302.    $1.90  net. 


FROM  THE  AUTHORS   PREFACE. 

At  the  present  day,  when  students  are  required  to  gain  knowledge  of  natural  phe- 
nomena  by  performing  experiments  for  themselves  in  laboratories,  every  teacher  rinds 
that  as  his  classes  increase  in  number,  some  difficulty  is  experienced  in  providing, 
during  a  limited  time,  ample  instruction  in  the  matter  of  details  and  methods. 

During  the  past  few  years  we  ourselves  have  had  such  difficulties  with  large  classes; 
and  that  is  our  reason  for  the  appearance  of  the  present  work,  which  is  the  natural 
outcome  of  our  experience.  We  know  that  it  will  be  of  service  to  our  own  students, 
and  hope  that  it  will  be  appreciated  by  those  engaged  in  teaching  Experimental 
Physics  elsewhere. 

The  book  contains  a  series  of  elementary  experiments  specially  adapted  for  stu- 
dents who  have  had  but  little  acquaintance  with  higher  mathematical  methods :  these 
are  arranged,  as  far  as  possible,  in  order  of  difficulty.  There  is  also  an  advanced 
course  of  experimental  work  in  Acoustics,  Heat,  and  Electricity  and  Magnetism, 
which  is  intended  for  those  who  have  taken  the  elementary  course. 

The  experiments  in  Acoustics  are  simple,  and  of  such  a  nature  that  the  most  of 
them  can  be  performed  by  beginners  in  the  study  of  Physics;  those  in  Heat,  although 
not  requiring  more  than  an  ordinary  acquaintance  with  Arithmetic,  are  more  tedious 
and  apt  to  test  the  patience  of  the  experimenter;  while  the  course  in  Electricity  and 
Magnetism  has  been  arranged  to  illustrate  the  fundamental  laws  of  the  mathematical 
theory,  and  involves  a  good  working  knowledge  of  the  Calculus. 


THE    MACMILLAN   COMPANY, 

66    FIFTH   AVENUE,  NEW  YORK. 


OF  THK 

TERSITT 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed 


below. 

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YC  32681 


